:: ROBBINS2 semantic presentation
:: deftheorem Def1 defines satisfying_DN_1 ROBBINS2:def 1 :
theorem Th1: :: ROBBINS2:1
theorem Th2: :: ROBBINS2:2
theorem Th3: :: ROBBINS2:3
theorem Th4: :: ROBBINS2:4
theorem Th5: :: ROBBINS2:5
theorem Th6: :: ROBBINS2:6
theorem Th7: :: ROBBINS2:7
theorem Th8: :: ROBBINS2:8
theorem Th9: :: ROBBINS2:9
theorem Th10: :: ROBBINS2:10
theorem Th11: :: ROBBINS2:11
theorem Th12: :: ROBBINS2:12
theorem Th13: :: ROBBINS2:13
theorem Th14: :: ROBBINS2:14
theorem Th15: :: ROBBINS2:15
theorem Th16: :: ROBBINS2:16
theorem Th17: :: ROBBINS2:17
theorem Th18: :: ROBBINS2:18
theorem Th19: :: ROBBINS2:19
theorem Th20: :: ROBBINS2:20
theorem Th21: :: ROBBINS2:21
theorem Th22: :: ROBBINS2:22
theorem Th23: :: ROBBINS2:23
theorem Th24: :: ROBBINS2:24
theorem Th25: :: ROBBINS2:25
theorem Th26: :: ROBBINS2:26
theorem Th27: :: ROBBINS2:27
Lemma29:
for b1 being non empty satisfying_DN_1 ComplLattStr holds b1 is join-commutative
theorem Th28: :: ROBBINS2:28
theorem Th29: :: ROBBINS2:29
theorem Th30: :: ROBBINS2:30
theorem Th31: :: ROBBINS2:31
theorem Th32: :: ROBBINS2:32
theorem Th33: :: ROBBINS2:33
theorem Th34: :: ROBBINS2:34
theorem Th35: :: ROBBINS2:35
theorem Th36: :: ROBBINS2:36
theorem Th37: :: ROBBINS2:37
theorem Th38: :: ROBBINS2:38
theorem Th39: :: ROBBINS2:39
theorem Th40: :: ROBBINS2:40
theorem Th41: :: ROBBINS2:41
theorem Th42: :: ROBBINS2:42
theorem Th43: :: ROBBINS2:43
theorem Th44: :: ROBBINS2:44
theorem Th45: :: ROBBINS2:45
theorem Th46: :: ROBBINS2:46
theorem Th47: :: ROBBINS2:47
theorem Th48: :: ROBBINS2:48
theorem Th49: :: ROBBINS2:49
theorem Th50: :: ROBBINS2:50
theorem Th51: :: ROBBINS2:51
theorem Th52: :: ROBBINS2:52
theorem Th53: :: ROBBINS2:53
theorem Th54: :: ROBBINS2:54
theorem Th55: :: ROBBINS2:55
theorem Th56: :: ROBBINS2:56
theorem Th57: :: ROBBINS2:57
Lemma53:
for b1 being non empty satisfying_DN_1 ComplLattStr holds b1 is join-associative
Lemma54:
for b1 being non empty satisfying_DN_1 ComplLattStr holds b1 is Robbins
theorem Th58: :: ROBBINS2:58
theorem Th59: :: ROBBINS2:59
:: deftheorem Def2 defines satisfying_MD_1 ROBBINS2:def 2 :
:: deftheorem Def3 defines satisfying_MD_2 ROBBINS2:def 3 :
E59:
now
let c
1 be non
empty ComplLattStr ;
assume E60:
( c
1 is
satisfying_MD_1 & c
1 is
satisfying_MD_2 )
;
E61:
for b
1, b
2 being
Element of c
1 holds
(b1 ` ) + ((b1 ` ) + b2) = (b1 ` ) + b
2
E62:
for b
1, b
2, b
3 being
Element of c
1 holds
(((((b1 ` ) + b2) ` ) + b3) ` ) + ((b1 ` ) + b3) = b
3 + ((b1 ` ) + b2)
proof
let c
2, c
3, c
4 be
Element of c
1;
(((((c2 ` ) + c3) ` ) + c4) ` ) + ((c2 ` ) + c4) =
c
4 + ((c2 ` ) + ((c2 ` ) + c3))
by E60, Def3
.=
c
4 + ((c2 ` ) + c3)
by E61
;
hence
(((((c2 ` ) + c3) ` ) + c4) ` ) + ((c2 ` ) + c4) = c
4 + ((c2 ` ) + c3)
;
end;
E63:
for b
1, b
2, b
3 being
Element of c
1 holds
(((b1 ` ) + b2) ` ) + ((((b1 ` ) + b3) ` ) + b2) = b
2 + b
1
E64:
for b
1, b
2, b
3 being
Element of c
1 holds
(((b1 ` ) + ((((b2 + b1) ` ) + b3) ` )) ` ) + (b2 + ((((b2 + b1) ` ) + b3) ` )) = b
2 + b
1
proof
let c
2, c
3, c
4 be
Element of c
1;
set c
5 =
(((c3 + c2) ` ) + c4) ` ;
(((c2 ` ) + ((((c3 + c2) ` ) + c4) ` )) ` ) + (c3 + ((((c3 + c2) ` ) + c4) ` )) =
((((c3 + c2) ` ) + c4) ` ) + (c3 + c2)
by E60, Def3
.=
c
3 + c
2
by E60, Def2
;
hence
(((c2 ` ) + ((((c3 + c2) ` ) + c4) ` )) ` ) + (c3 + ((((c3 + c2) ` ) + c4) ` )) = c
3 + c
2
;
end;
E65:
for b
1, b
2 being
Element of c
1 holds
(((b1 ` ) + b2) ` ) + b
2 = b
2 + b
1
E66:
for b
1, b
2 being
Element of c
1 holds b
1 + (b2 + (b2 + b1)) = b
2 + b
1
E67:
for b
1, b
2 being
Element of c
1 holds b
1 + ((b2 ` ) + b1) = (b2 ` ) + b
1
E68:
for b
1 being
Element of c
1 holds b
1 + b
1 = b
1
E69:
for b
1, b
2 being
Element of c
1 holds b
1 + (b1 + b2) = b
1 + b
2
E70:
for b
1, b
2 being
Element of c
1 holds
(b1 + b2) + b
2 = b
1 + b
2
proof
let c
2, c
3 be
Element of c
1;
set c
4 = c
2 + c
3;
(c2 + c3) + c
3 =
(((c3 ` ) + (c2 + c3)) ` ) + (c2 + c3)
by E65
.=
(((c3 ` ) + (c2 + c3)) ` ) + (c2 + (c2 + c3))
by E69
.=
(c2 + c3) + (c2 + c3)
by E60, Def3
.=
c
2 + c
3
by E68
;
hence
(c2 + c3) + c
3 = c
2 + c
3
;
end;
E71:
for b
1 being
Element of c
1 holds
((b1 ` ) ` ) + b
1 = b
1
E72:
for b
1, b
2 being
Element of c
1 holds b
1 + (b2 + b1) = b
2 + b
1
E73:
for b
1, b
2 being
Element of c
1 holds b
1 + (((b1 ` ) ` ) + b2) = b
1 + b
2
E74:
for b
1, b
2 being
Element of c
1 holds
(b1 + b2) + b
1 = b
1 + b
2
E75:
for b
1, b
2, b
3 being
Element of c
1 holds
(b1 + b2) + (b2 + b3) = (b1 + b2) + b
3
proof
let c
2, c
3, c
4 be
Element of c
1;
set c
5 = c
2 + c
3;
(c2 + c3) + c
4 =
(((c4 ` ) + (c2 + c3)) ` ) + (c2 + c3)
by E65
.=
(((c4 ` ) + (c2 + c3)) ` ) + (c3 + (c2 + c3))
by E72
.=
(c2 + c3) + (c3 + c4)
by E60, Def3
;
hence
(c2 + c3) + (c3 + c4) = (c2 + c3) + c
4
;
end;
E76:
for b
1, b
2 being
Element of c
1 holds
((b1 + (b2 ` )) ` ) + b
2 = b
2
E77:
for b
1 being
Element of c
1 holds b
1 + ((b1 ` ) ` ) = b
1
E78:
for b
1, b
2 being
Element of c
1 holds b
1 + (((b1 ` ) + b2) ` ) = b
1
E79:
for b
1, b
2 being
Element of c
1 holds b
1 + ((b2 + (b1 ` )) ` ) = b
1
proof
let c
2, c
3 be
Element of c
1;
c
2 + ((c3 + (c2 ` )) ` ) =
(((c3 + (c2 ` )) ` ) + c2) + ((c3 + (c2 ` )) ` )
by E76
.=
((c3 + (c2 ` )) ` ) + c
2
by E74
.=
c
2
by E76
;
hence
c
2 + ((c3 + (c2 ` )) ` ) = c
2
;
end;
E80:
for b
1, b
2 being
Element of c
1 holds
(((b1 ` ) + ((b1 + b2) ` )) ` ) + ((b2 ` ) + ((b1 + b2) ` )) = (b2 ` ) + b
1
proof
let c
2, c
3 be
Element of c
1;
(((c2 ` ) + ((c2 + c3) ` )) ` ) + ((c3 ` ) + ((c2 + c3) ` )) =
(((c2 ` ) + ((c2 + c3) ` )) ` ) + ((c3 ` ) + (((((c3 ` ) + c2) ` ) + c2) ` ))
by E65
.=
(((c2 ` ) + (((((c3 ` ) + c2) ` ) + c2) ` )) ` ) + ((c3 ` ) + (((((c3 ` ) + c2) ` ) + c2) ` ))
by E65
.=
(c3 ` ) + c
2
by E64
;
hence
(((c2 ` ) + ((c2 + c3) ` )) ` ) + ((c3 ` ) + ((c2 + c3) ` )) = (c3 ` ) + c
2
;
end;
E81:
for b
1, b
2 being
Element of c
1 holds
((b1 + b2) ` ) + b
1 = (b2 ` ) + b
1
proof
let c
2, c
3 be
Element of c
1;
set c
4 =
(c3 ` ) + c
2;
(c3 ` ) + c
2 =
c
2 + ((c3 ` ) + c2)
by E67
.=
(((((c3 ` ) + c2) ` ) + c2) ` ) + c
2
by E65
.=
((c2 + c3) ` ) + c
2
by E65
;
hence
((c2 + c3) ` ) + c
2 = (c3 ` ) + c
2
;
end;
E82:
for b
1, b
2 being
Element of c
1 holds
((b1 + b2) ` ) + (((b2 ` ) + b1) ` ) = (b1 ` ) + (((b2 ` ) + b1) ` )
proof
let c
2, c
3 be
Element of c
1;
((c2 + c3) ` ) + (((c3 ` ) + c2) ` ) =
(((((c3 ` ) + c2) ` ) + c2) ` ) + (((c3 ` ) + c2) ` )
by E65
.=
(c2 ` ) + (((c3 ` ) + c2) ` )
by E81
;
hence
((c2 + c3) ` ) + (((c3 ` ) + c2) ` ) = (c2 ` ) + (((c3 ` ) + c2) ` )
;
end;
E83:
for b
1 being
Element of c
1 holds b
1 + ((((b1 ` ) ` ) ` ) ` ) = b
1
E84:
for b
1 being
Element of c
1 holds
((b1 ` ) ` ) ` = b
1 `
E85:
for b
1, b
2 being
Element of c
1 holds b
1 + ((b2 ` ) ` ) = b
1 + b
2
proof
let c
2, c
3 be
Element of c
1;
c
2 + c
3 =
(((c3 ` ) + c2) ` ) + ((((c3 ` ) + c2) ` ) + c2)
by E63
.=
(((c3 ` ) + c2) ` ) + ((((((c3 ` ) ` ) ` ) + c2) ` ) + c2)
by E84
.=
(((((c3 ` ) ` ) ` ) + c2) ` ) + ((((((c3 ` ) ` ) ` ) + c2) ` ) + c2)
by E84
.=
c
2 + ((c3 ` ) ` )
by E63
;
hence
c
2 + ((c3 ` ) ` ) = c
2 + c
3
;
end;
E86:
for b
1 being
Element of c
1 holds
(b1 ` ) ` = b
1
E87:
for b
1, b
2 being
Element of c
1 holds
(b1 ` ) + ((b2 + b1) ` ) = b
1 `
E88:
for b
1, b
2 being
Element of c
1 holds
(b1 ` ) + ((b1 + b2) ` ) = b
1 `
E89:
for b
1, b
2 being
Element of c
1 holds b
1 + (b2 ` ) = (b2 ` ) + b
1
E90:
for b
1, b
2 being
Element of c
1 holds b
1 + b
2 = b
2 + b
1
hence
c
1 is
join-commutative
by LATTICES:def 4;
for b
1, b
2 being
Element of c
1 holds
(((b1 ` ) + (b2 ` )) ` ) + (((b1 ` ) + b2) ` ) = b
1
proof
let c
2, c
3 be
Element of c
1;
(((c2 ` ) + (c3 ` )) ` ) + (((c2 ` ) + c3) ` ) =
(((c3 ` ) + (c2 ` )) ` ) + (((c2 ` ) + c3) ` )
by E90
.=
(((c2 ` ) + c3) ` ) + (((c3 ` ) + (c2 ` )) ` )
by E90
.=
((c2 ` ) ` ) + (((c3 ` ) + (c2 ` )) ` )
by E82
.=
c
2 + (((c3 ` ) + (c2 ` )) ` )
by E86
.=
c
2
by E79
;
hence
(((c2 ` ) + (c3 ` )) ` ) + (((c2 ` ) + c3) ` ) = c
2
;
end;
hence
c
1 is
Huntington
by ROBBINS1:def 6;
for b
1, b
2, b
3 being
Element of c
1 holds
(b1 + b2) + b
3 = b
1 + (b2 + b3)
proof
let c
2, c
3, c
4 be
Element of c
1;
for b
1, b
2, b
3 being
Element of c
1 holds
(b1 + b2) + (b3 + b1) = b
2 + (b1 + b3)
proof
let c
5, c
6, c
7 be
Element of c
1;
(c5 + c6) + (c7 + c5) =
(c7 + c5) + (c5 + c6)
by E90
.=
(c7 + c5) + c
6
by E75
.=
(c5 + c7) + c
6
by E90
.=
c
6 + (c5 + c7)
by E90
;
hence
(c5 + c6) + (c7 + c5) = c
6 + (c5 + c7)
;
end;
then
(c3 + c2) + (c4 + c3) = c
2 + (c3 + c4)
;
then E91:
(c2 + c3) + (c4 + c3) = c
2 + (c3 + c4)
by E90;
(c2 + c3) + c
4 =
(c2 + c3) + (c3 + c4)
by E75
.=
c
2 + (c3 + c4)
by E90, E91
;
hence
(c2 + c3) + c
4 = c
2 + (c3 + c4)
;
end;
hence
c
1 is
join-associative
by LATTICES:def 5;
end;