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Theorem stoweidlem23 29823
Description: This lemma is used to prove the existence of a function pt as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem23.1  |-  F/ t
ph
stoweidlem23.2  |-  F/_ t G
stoweidlem23.3  |-  H  =  ( t  e.  T  |->  ( ( G `  t )  -  ( G `  Z )
) )
stoweidlem23.4  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem23.5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem23.6  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweidlem23.7  |-  ( ph  ->  S  e.  T )
stoweidlem23.8  |-  ( ph  ->  Z  e.  T )
stoweidlem23.9  |-  ( ph  ->  G  e.  A )
stoweidlem23.10  |-  ( ph  ->  ( G `  S
)  =/=  ( G `
 Z ) )
Assertion
Ref Expression
stoweidlem23  |-  ( ph  ->  ( H  e.  A  /\  ( H `  S
)  =/=  ( H `
 Z )  /\  ( H `  Z )  =  0 ) )
Distinct variable groups:    f, g,
t, T    A, f,
g    f, G, g    ph, f,
g    g, Z, t    x, t, T    t, S    x, A    x, G    x, Z    ph, x
Allowed substitution hints:    ph( t)    A( t)    S( x, f, g)    G( t)    H( x, t, f, g)    Z( f)

Proof of Theorem stoweidlem23
StepHypRef Expression
1 stoweidlem23.3 . . 3  |-  H  =  ( t  e.  T  |->  ( ( G `  t )  -  ( G `  Z )
) )
2 stoweidlem23.1 . . . . 5  |-  F/ t
ph
3 stoweidlem23.9 . . . . . . . . 9  |-  ( ph  ->  G  e.  A )
43ancli 551 . . . . . . . . 9  |-  ( ph  ->  ( ph  /\  G  e.  A ) )
5 eleq1 2503 . . . . . . . . . . . 12  |-  ( f  =  G  ->  (
f  e.  A  <->  G  e.  A ) )
65anbi2d 703 . . . . . . . . . . 11  |-  ( f  =  G  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  G  e.  A ) ) )
7 feq1 5547 . . . . . . . . . . 11  |-  ( f  =  G  ->  (
f : T --> RR  <->  G : T
--> RR ) )
86, 7imbi12d 320 . . . . . . . . . 10  |-  ( f  =  G  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  G  e.  A )  ->  G : T --> RR ) ) )
9 stoweidlem23.4 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
108, 9vtoclg 3035 . . . . . . . . 9  |-  ( G  e.  A  ->  (
( ph  /\  G  e.  A )  ->  G : T --> RR ) )
113, 4, 10sylc 60 . . . . . . . 8  |-  ( ph  ->  G : T --> RR )
1211fnvinran 29741 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  e.  RR )
1312recnd 9417 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  e.  CC )
14 stoweidlem23.8 . . . . . . . . 9  |-  ( ph  ->  Z  e.  T )
1511, 14ffvelrnd 5849 . . . . . . . 8  |-  ( ph  ->  ( G `  Z
)  e.  RR )
1615adantr 465 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  Z )  e.  RR )
1716recnd 9417 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  Z )  e.  CC )
1813, 17negsubd 9730 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  +  -u ( G `  Z )
)  =  ( ( G `  t )  -  ( G `  Z ) ) )
192, 18mpteq2da 4382 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  -u ( G `  Z ) ) )  =  ( t  e.  T  |->  ( ( G `  t
)  -  ( G `
 Z ) ) ) )
20 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
2115renegcld 9780 . . . . . . . . 9  |-  ( ph  -> 
-u ( G `  Z )  e.  RR )
2221adantr 465 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  -u ( G `  Z )  e.  RR )
23 eqid 2443 . . . . . . . . 9  |-  ( t  e.  T  |->  -u ( G `  Z )
)  =  ( t  e.  T  |->  -u ( G `  Z )
)
2423fvmpt2 5786 . . . . . . . 8  |-  ( ( t  e.  T  /\  -u ( G `  Z
)  e.  RR )  ->  ( ( t  e.  T  |->  -u ( G `  Z )
) `  t )  =  -u ( G `  Z ) )
2520, 22, 24syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( t  e.  T  |-> 
-u ( G `  Z ) ) `  t )  =  -u ( G `  Z ) )
2625oveq2d 6112 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  +  ( ( t  e.  T  |->  -u ( G `  Z ) ) `  t ) )  =  ( ( G `  t )  +  -u ( G `  Z ) ) )
272, 26mpteq2da 4382 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  ( ( t  e.  T  |-> 
-u ( G `  Z ) ) `  t ) ) )  =  ( t  e.  T  |->  ( ( G `
 t )  + 
-u ( G `  Z ) ) ) )
2821ancli 551 . . . . . . 7  |-  ( ph  ->  ( ph  /\  -u ( G `  Z )  e.  RR ) )
29 eleq1 2503 . . . . . . . . . 10  |-  ( x  =  -u ( G `  Z )  ->  (
x  e.  RR  <->  -u ( G `
 Z )  e.  RR ) )
3029anbi2d 703 . . . . . . . . 9  |-  ( x  =  -u ( G `  Z )  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  -u ( G `  Z
)  e.  RR ) ) )
31 stoweidlem23.2 . . . . . . . . . . . . . 14  |-  F/_ t G
32 nfcv 2584 . . . . . . . . . . . . . 14  |-  F/_ t Z
3331, 32nffv 5703 . . . . . . . . . . . . 13  |-  F/_ t
( G `  Z
)
3433nfneg 9611 . . . . . . . . . . . 12  |-  F/_ t -u ( G `  Z
)
3534nfeq2 2595 . . . . . . . . . . 11  |-  F/ t  x  =  -u ( G `  Z )
36 simpl 457 . . . . . . . . . . 11  |-  ( ( x  =  -u ( G `  Z )  /\  t  e.  T
)  ->  x  =  -u ( G `  Z
) )
3735, 36mpteq2da 4382 . . . . . . . . . 10  |-  ( x  =  -u ( G `  Z )  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  -u ( G `  Z )
) )
3837eleq1d 2509 . . . . . . . . 9  |-  ( x  =  -u ( G `  Z )  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  -u ( G `  Z ) )  e.  A ) )
3930, 38imbi12d 320 . . . . . . . 8  |-  ( x  =  -u ( G `  Z )  ->  (
( ( ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A
)  <->  ( ( ph  /\  -u ( G `  Z
)  e.  RR )  ->  ( t  e.  T  |->  -u ( G `  Z ) )  e.  A ) ) )
40 stoweidlem23.6 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
4139, 40vtoclg 3035 . . . . . . 7  |-  ( -u ( G `  Z )  e.  RR  ->  (
( ph  /\  -u ( G `  Z )  e.  RR )  ->  (
t  e.  T  |->  -u ( G `  Z ) )  e.  A ) )
4221, 28, 41sylc 60 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |-> 
-u ( G `  Z ) )  e.  A )
43 stoweidlem23.5 . . . . . . 7  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
44 nfmpt1 4386 . . . . . . 7  |-  F/_ t
( t  e.  T  |-> 
-u ( G `  Z ) )
4543, 31, 44stoweidlem8 29808 . . . . . 6  |-  ( (
ph  /\  G  e.  A  /\  ( t  e.  T  |->  -u ( G `  Z ) )  e.  A )  ->  (
t  e.  T  |->  ( ( G `  t
)  +  ( ( t  e.  T  |->  -u ( G `  Z ) ) `  t ) ) )  e.  A
)
463, 42, 45mpd3an23 1316 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  ( ( t  e.  T  |-> 
-u ( G `  Z ) ) `  t ) ) )  e.  A )
4727, 46eqeltrrd 2518 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  -u ( G `  Z ) ) )  e.  A
)
4819, 47eqeltrrd 2518 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( G `  Z )
) )  e.  A
)
491, 48syl5eqel 2527 . 2  |-  ( ph  ->  H  e.  A )
50 stoweidlem23.7 . . . . . 6  |-  ( ph  ->  S  e.  T )
5111, 50ffvelrnd 5849 . . . . 5  |-  ( ph  ->  ( G `  S
)  e.  RR )
5251recnd 9417 . . . 4  |-  ( ph  ->  ( G `  S
)  e.  CC )
5315recnd 9417 . . . 4  |-  ( ph  ->  ( G `  Z
)  e.  CC )
54 stoweidlem23.10 . . . 4  |-  ( ph  ->  ( G `  S
)  =/=  ( G `
 Z ) )
5552, 53, 54subne0d 9733 . . 3  |-  ( ph  ->  ( ( G `  S )  -  ( G `  Z )
)  =/=  0 )
5651, 15resubcld 9781 . . . 4  |-  ( ph  ->  ( ( G `  S )  -  ( G `  Z )
)  e.  RR )
57 nfcv 2584 . . . . 5  |-  F/_ t S
5831, 57nffv 5703 . . . . . 6  |-  F/_ t
( G `  S
)
59 nfcv 2584 . . . . . 6  |-  F/_ t  -
6058, 59, 33nfov 6119 . . . . 5  |-  F/_ t
( ( G `  S )  -  ( G `  Z )
)
61 fveq2 5696 . . . . . 6  |-  ( t  =  S  ->  ( G `  t )  =  ( G `  S ) )
6261oveq1d 6111 . . . . 5  |-  ( t  =  S  ->  (
( G `  t
)  -  ( G `
 Z ) )  =  ( ( G `
 S )  -  ( G `  Z ) ) )
6357, 60, 62, 1fvmptf 5795 . . . 4  |-  ( ( S  e.  T  /\  ( ( G `  S )  -  ( G `  Z )
)  e.  RR )  ->  ( H `  S )  =  ( ( G `  S
)  -  ( G `
 Z ) ) )
6450, 56, 63syl2anc 661 . . 3  |-  ( ph  ->  ( H `  S
)  =  ( ( G `  S )  -  ( G `  Z ) ) )
6515, 15resubcld 9781 . . . . 5  |-  ( ph  ->  ( ( G `  Z )  -  ( G `  Z )
)  e.  RR )
6633, 59, 33nfov 6119 . . . . . 6  |-  F/_ t
( ( G `  Z )  -  ( G `  Z )
)
67 fveq2 5696 . . . . . . 7  |-  ( t  =  Z  ->  ( G `  t )  =  ( G `  Z ) )
6867oveq1d 6111 . . . . . 6  |-  ( t  =  Z  ->  (
( G `  t
)  -  ( G `
 Z ) )  =  ( ( G `
 Z )  -  ( G `  Z ) ) )
6932, 66, 68, 1fvmptf 5795 . . . . 5  |-  ( ( Z  e.  T  /\  ( ( G `  Z )  -  ( G `  Z )
)  e.  RR )  ->  ( H `  Z )  =  ( ( G `  Z
)  -  ( G `
 Z ) ) )
7014, 65, 69syl2anc 661 . . . 4  |-  ( ph  ->  ( H `  Z
)  =  ( ( G `  Z )  -  ( G `  Z ) ) )
7153subidd 9712 . . . 4  |-  ( ph  ->  ( ( G `  Z )  -  ( G `  Z )
)  =  0 )
7270, 71eqtrd 2475 . . 3  |-  ( ph  ->  ( H `  Z
)  =  0 )
7355, 64, 723netr4d 2640 . 2  |-  ( ph  ->  ( H `  S
)  =/=  ( H `
 Z ) )
7449, 73, 723jca 1168 1  |-  ( ph  ->  ( H  e.  A  /\  ( H `  S
)  =/=  ( H `
 Z )  /\  ( H `  Z )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2571    =/= wne 2611    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287    + caddc 9290    - cmin 9600   -ucneg 9601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-sub 9602  df-neg 9603
This theorem is referenced by:  stoweidlem43  29843
  Copyright terms: Public domain W3C validator