:: SUPINF_2 semantic presentation
:: deftheorem Def1 defines 0. SUPINF_2:def 1 :
definition
let x,
y be
R_eal;
func x + y -> R_eal means :
Def2:
:: SUPINF_2:def 2
ex
a,
b being
Real st
(
x = a &
y = b &
it = a + b )
if (
x in REAL &
y in REAL )
it = +infty if ( (
x = +infty &
y <> -infty ) or (
y = +infty &
x <> -infty ) )
it = -infty if ( (
x = -infty &
y <> +infty ) or (
y = -infty &
x <> +infty ) )
otherwise it = 0. ;
existence
( ( x in REAL & y in REAL implies ex b1 being R_eal ex a, b being Real st
( x = a & y = b & b1 = a + b ) ) & ( ( ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) ) implies ex b1 being R_eal st b1 = +infty ) & ( ( ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) ) implies ex b1 being R_eal st b1 = -infty ) & ( ( x in REAL & y in REAL ) or ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) or ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) or ex b1 being R_eal st b1 = 0. ) )
uniqueness
for b1, b2 being R_eal holds
( ( x in REAL & y in REAL & ex a, b being Real st
( x = a & y = b & b1 = a + b ) & ex a, b being Real st
( x = a & y = b & b2 = a + b ) implies b1 = b2 ) & ( ( ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) ) & b1 = +infty & b2 = +infty implies b1 = b2 ) & ( ( ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) ) & b1 = -infty & b2 = -infty implies b1 = b2 ) & ( ( x in REAL & y in REAL ) or ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) or ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) or not b1 = 0. or not b2 = 0. or b1 = b2 ) )
;
consistency
for b1 being R_eal holds
( ( x in REAL & y in REAL & ( ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) ) implies ( ex a, b being Real st
( x = a & y = b & b1 = a + b ) iff b1 = +infty ) ) & ( x in REAL & y in REAL & ( ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) ) implies ( ex a, b being Real st
( x = a & y = b & b1 = a + b ) iff b1 = -infty ) ) & ( ( ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) ) & ( ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) ) implies ( b1 = +infty iff b1 = -infty ) ) )
by XREAL_0:def 1;
commutativity
for b1, x, y being R_eal st ( x in REAL & y in REAL implies ex a, b being Real st
( x = a & y = b & b1 = a + b ) ) & ( ( ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) ) implies b1 = +infty ) & ( ( ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) ) implies b1 = -infty ) & ( ( x in REAL & y in REAL ) or ( x = +infty & y <> -infty ) or ( y = +infty & x <> -infty ) or ( x = -infty & y <> +infty ) or ( y = -infty & x <> +infty ) or b1 = 0. ) holds
( ( y in REAL & x in REAL implies ex a, b being Real st
( y = a & x = b & b1 = a + b ) ) & ( ( ( y = +infty & x <> -infty ) or ( x = +infty & y <> -infty ) ) implies b1 = +infty ) & ( ( ( y = -infty & x <> +infty ) or ( x = -infty & y <> +infty ) ) implies b1 = -infty ) & ( ( y in REAL & x in REAL ) or ( y = +infty & x <> -infty ) or ( x = +infty & y <> -infty ) or ( y = -infty & x <> +infty ) or ( x = -infty & y <> +infty ) or b1 = 0. ) )
;
end;
:: deftheorem Def2 defines + SUPINF_2:def 2 :
theorem Th1: :: SUPINF_2:1
for
x,
y being
R_eal for
a,
b being
Real st
x = a &
y = b holds
x + y = a + b
theorem Th2: :: SUPINF_2:2
:: deftheorem Def3 defines - SUPINF_2:def 3 :
:: deftheorem defines - SUPINF_2:def 4 :
for
x,
y being
R_eal holds
x - y = x + (- y);
theorem Th3: :: SUPINF_2:3
Lm1:
( not +infty in REAL & not -infty in REAL )
by XREAL_0:def 1;
theorem Th4: :: SUPINF_2:4
theorem Th5: :: SUPINF_2:5
for
x,
y being
R_eal for
a,
b being
Real st
x = a &
y = b holds
x - y = a - b
theorem Th6: :: SUPINF_2:6
theorem Th7: :: SUPINF_2:7
theorem Th8: :: SUPINF_2:8
theorem Th9: :: SUPINF_2:9
theorem Th10: :: SUPINF_2:10
theorem Th11: :: SUPINF_2:11
theorem Th12: :: SUPINF_2:12
theorem Th13: :: SUPINF_2:13
theorem Th14: :: SUPINF_2:14
theorem :: SUPINF_2:15
Lm2:
for x being R_eal holds
( - x in REAL iff x in REAL )
Lm3:
for x, y being R_eal st x <= y holds
- y <= - x
theorem Th16: :: SUPINF_2:16
theorem :: SUPINF_2:17
theorem Th18: :: SUPINF_2:18
theorem :: SUPINF_2:19
theorem Th20: :: SUPINF_2:20
Lm4:
for x, y, s, t being R_eal st 0. <= x & 0. <= s & x <= y & s <= t holds
x + s <= y + t
theorem Th21: :: SUPINF_2:21
:: deftheorem Def5 defines + SUPINF_2:def 5 :
:: deftheorem Def6 defines - SUPINF_2:def 6 :
theorem Th22: :: SUPINF_2:22
theorem Th23: :: SUPINF_2:23
theorem :: SUPINF_2:24
theorem :: SUPINF_2:25
theorem Th26: :: SUPINF_2:26
theorem Th27: :: SUPINF_2:27
theorem :: SUPINF_2:28
theorem :: SUPINF_2:29
theorem Th30: :: SUPINF_2:30
theorem Th31: :: SUPINF_2:31
theorem Th32: :: SUPINF_2:32
theorem Th33: :: SUPINF_2:33
:: deftheorem defines sup SUPINF_2:def 7 :
:: deftheorem defines inf SUPINF_2:def 8 :
:: deftheorem Def9 defines + SUPINF_2:def 9 :
theorem Th34: :: SUPINF_2:34
theorem Th35: :: SUPINF_2:35
theorem Th36: :: SUPINF_2:36
:: deftheorem Def10 defines - SUPINF_2:def 10 :
theorem Th37: :: SUPINF_2:37
theorem Th38: :: SUPINF_2:38
:: deftheorem Def11 defines bounded_above SUPINF_2:def 11 :
:: deftheorem Def12 defines bounded_below SUPINF_2:def 12 :
:: deftheorem Def13 defines bounded SUPINF_2:def 13 :
theorem :: SUPINF_2:39
theorem Th40: :: SUPINF_2:40
theorem Th41: :: SUPINF_2:41
theorem :: SUPINF_2:42
theorem :: SUPINF_2:43
theorem :: SUPINF_2:44
theorem Th45: :: SUPINF_2:45
theorem Th46: :: SUPINF_2:46
theorem Th47: :: SUPINF_2:47
theorem :: SUPINF_2:48
theorem Th49: :: SUPINF_2:49
theorem Th50: :: SUPINF_2:50
theorem :: SUPINF_2:51
theorem Th52: :: SUPINF_2:52
:: deftheorem Def14 defines denumerable SUPINF_2:def 14 :
:: deftheorem Def15 defines nonnegative SUPINF_2:def 15 :
:: deftheorem Def16 defines Num SUPINF_2:def 16 :
theorem Th53: :: SUPINF_2:53
:: deftheorem Def17 defines Ser SUPINF_2:def 17 :
theorem Th54: :: SUPINF_2:54
theorem Th55: :: SUPINF_2:55
theorem Th56: :: SUPINF_2:56
:: deftheorem defines Set_of_Series SUPINF_2:def 18 :
Lm5:
for F being Function of NAT , ExtREAL holds rng F is non empty Subset of ExtREAL
:: deftheorem defines SUM SUPINF_2:def 19 :
:: deftheorem defines is_sumable SUPINF_2:def 20 :
theorem Th57: :: SUPINF_2:57
:: deftheorem Def21 defines Ser SUPINF_2:def 21 :
:: deftheorem Def22 defines nonnegative SUPINF_2:def 22 :
:: deftheorem defines SUM SUPINF_2:def 23 :
theorem Th58: :: SUPINF_2:58
theorem Th59: :: SUPINF_2:59
theorem Th60: :: SUPINF_2:60
theorem Th61: :: SUPINF_2:61
theorem Th62: :: SUPINF_2:62
theorem Th63: :: SUPINF_2:63
theorem Th64: :: SUPINF_2:64
:: deftheorem Def24 defines summable SUPINF_2:def 24 :
theorem :: SUPINF_2:65
theorem :: SUPINF_2:66
theorem Th67: :: SUPINF_2:67
theorem Th68: :: SUPINF_2:68
theorem :: SUPINF_2:69
theorem :: SUPINF_2:70
theorem :: SUPINF_2:71