:: WAYBEL_7 semantic presentation
theorem Th1: :: WAYBEL_7:1
canceled;
theorem Th2: :: WAYBEL_7:2
canceled;
theorem Th3: :: WAYBEL_7:3
theorem Th4: :: WAYBEL_7:4
theorem Th5: :: WAYBEL_7:5
theorem Th6: :: WAYBEL_7:6
canceled;
theorem Th7: :: WAYBEL_7:7
canceled;
theorem Th8: :: WAYBEL_7:8
theorem Th9: :: WAYBEL_7:9
theorem Th10: :: WAYBEL_7:10
theorem Th11: :: WAYBEL_7:11
theorem Th12: :: WAYBEL_7:12
theorem Th13: :: WAYBEL_7:13
theorem Th14: :: WAYBEL_7:14
theorem Th15: :: WAYBEL_7:15
:: deftheorem Def1 defines prime WAYBEL_7:def 1 :
theorem Th16: :: WAYBEL_7:16
theorem Th17: :: WAYBEL_7:17
:: deftheorem Def2 defines prime WAYBEL_7:def 2 :
theorem Th18: :: WAYBEL_7:18
theorem Th19: :: WAYBEL_7:19
theorem Th20: :: WAYBEL_7:20
theorem Th21: :: WAYBEL_7:21
theorem Th22: :: WAYBEL_7:22
theorem Th23: :: WAYBEL_7:23
theorem Th24: :: WAYBEL_7:24
theorem Th25: :: WAYBEL_7:25
:: deftheorem Def3 defines ultra WAYBEL_7:def 3 :
Lemma18:
for b1 being with_infima Poset
for b2 being Filter of b1
for b3 being non empty finite Subset of b1
for b4 being Element of b1 holds
not ( b4 in uparrow (fininfs (b2 \/ b3)) & ( for b5 being Element of b1 holds
not ( b5 in b2 & b4 >= b5 "/\" (inf b3) ) ) )
theorem Th26: :: WAYBEL_7:26
Lemma20:
for b1 being with_suprema Poset
for b2 being Ideal of b1
for b3 being non empty finite Subset of b1
for b4 being Element of b1 holds
not ( b4 in downarrow (finsups (b2 \/ b3)) & ( for b5 being Element of b1 holds
not ( b5 in b2 & b4 <= b5 "\/" (sup b3) ) ) )
theorem Th27: :: WAYBEL_7:27
theorem Th28: :: WAYBEL_7:28
theorem Th29: :: WAYBEL_7:29
theorem Th30: :: WAYBEL_7:30
:: deftheorem Def4 defines is_a_cluster_point_of WAYBEL_7:def 4 :
:: deftheorem Def5 defines is_a_convergence_point_of WAYBEL_7:def 5 :
theorem Th31: :: WAYBEL_7:31
theorem Th32: :: WAYBEL_7:32
theorem Th33: :: WAYBEL_7:33
theorem Th34: :: WAYBEL_7:34
theorem Th35: :: WAYBEL_7:35
theorem Th36: :: WAYBEL_7:36
theorem Th37: :: WAYBEL_7:37
:: deftheorem Def6 defines pseudoprime WAYBEL_7:def 6 :
theorem Th38: :: WAYBEL_7:38
theorem Th39: :: WAYBEL_7:39
theorem Th40: :: WAYBEL_7:40
theorem Th41: :: WAYBEL_7:41
theorem Th42: :: WAYBEL_7:42
theorem Th43: :: WAYBEL_7:43
:: deftheorem Def7 defines multiplicative WAYBEL_7:def 7 :
theorem Th44: :: WAYBEL_7:44
theorem Th45: :: WAYBEL_7:45
theorem Th46: :: WAYBEL_7:46
theorem Th47: :: WAYBEL_7:47
E36:
now
let c
1 be
lower-bounded continuous LATTICE;
let c
2 be
Element of c
1;
assume that E37:
c
1 -waybelow is
multiplicative
and E38:
for b
1, b
2 being
Element of c
1 holds
not ( b
1 "/\" b
2 << c
2 & not b
1 <= c
2 & not b
2 <= c
2 )
and E39:
not c
2 is
prime
;
consider c
3, c
4 being
Element of c
1 such that E40:
( c
3 "/\" c
4 <= c
2 & not c
3 <= c
2 & not c
4 <= c
2 )
by E39, WAYBEL_6:def 6;
E41:
for b
1 being
Element of c
1 holds
( not
waybelow b
1 is
empty &
waybelow b
1 is
directed )
;
then consider c
5 being
Element of c
1 such that E42:
( c
5 << c
3 & not c
5 <= c
2 )
by E40, WAYBEL_3:24;
consider c
6 being
Element of c
1 such that E43:
( c
6 << c
4 & not c
6 <= c
2 )
by E40, E41, WAYBEL_3:24;
(
[c5,c3] in c
1 -waybelow &
[c6,c4] in c
1 -waybelow )
by E42, E43, WAYBEL_4:def 2;
then
[(c5 "/\" c6),(c3 "/\" c4)] in c
1 -waybelow
by E37, Th45;
then
c
5 "/\" c
6 << c
3 "/\" c
4
by WAYBEL_4:def 2;
then
c
5 "/\" c
6 << c
2
by E40, WAYBEL_3:2;
hence
not verum
by E38, E42, E43;
end;
theorem Th48: :: WAYBEL_7:48
theorem Th49: :: WAYBEL_7:49
theorem Th50: :: WAYBEL_7:50