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Theorem stoweidlem23 27762
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 536 . . . . . . . . 9  |-  ( ph  ->  ( ph  /\  G  e.  A ) )
5 eleq1 2498 . . . . . . . . . . . 12  |-  ( f  =  G  ->  (
f  e.  A  <->  G  e.  A ) )
65anbi2d 686 . . . . . . . . . . 11  |-  ( f  =  G  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  G  e.  A ) ) )
7 feq1 5579 . . . . . . . . . . 11  |-  ( f  =  G  ->  (
f : T --> RR  <->  G : T
--> RR ) )
86, 7imbi12d 313 . . . . . . . . . 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 3013 . . . . . . . . 9  |-  ( G  e.  A  ->  (
( ph  /\  G  e.  A )  ->  G : T --> RR ) )
113, 4, 10sylc 59 . . . . . . . 8  |-  ( ph  ->  G : T --> RR )
1211fnvinran 27675 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  e.  RR )
1312recnd 9119 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  e.  CC )
14 stoweidlem23.8 . . . . . . . . 9  |-  ( ph  ->  Z  e.  T )
1511, 14ffvelrnd 5874 . . . . . . . 8  |-  ( ph  ->  ( G `  Z
)  e.  RR )
1615adantr 453 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  Z )  e.  RR )
1716recnd 9119 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  Z )  e.  CC )
1813, 17negsubd 9422 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  +  -u ( G `  Z )
)  =  ( ( G `  t )  -  ( G `  Z ) ) )
192, 18mpteq2da 4297 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  -u ( G `  Z ) ) )  =  ( t  e.  T  |->  ( ( G `  t
)  -  ( G `
 Z ) ) ) )
20 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
2115renegcld 9469 . . . . . . . . 9  |-  ( ph  -> 
-u ( G `  Z )  e.  RR )
2221adantr 453 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  -u ( G `  Z )  e.  RR )
23 eqid 2438 . . . . . . . . 9  |-  ( t  e.  T  |->  -u ( G `  Z )
)  =  ( t  e.  T  |->  -u ( G `  Z )
)
2423fvmpt2 5815 . . . . . . . 8  |-  ( ( t  e.  T  /\  -u ( G `  Z
)  e.  RR )  ->  ( ( t  e.  T  |->  -u ( G `  Z )
) `  t )  =  -u ( G `  Z ) )
2520, 22, 24syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( t  e.  T  |-> 
-u ( G `  Z ) ) `  t )  =  -u ( G `  Z ) )
2625oveq2d 6100 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  +  ( ( t  e.  T  |->  -u ( G `  Z ) ) `  t ) )  =  ( ( G `  t )  +  -u ( G `  Z ) ) )
272, 26mpteq2da 4297 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  ( ( t  e.  T  |-> 
-u ( G `  Z ) ) `  t ) ) )  =  ( t  e.  T  |->  ( ( G `
 t )  + 
-u ( G `  Z ) ) ) )
2821ancli 536 . . . . . . 7  |-  ( ph  ->  ( ph  /\  -u ( G `  Z )  e.  RR ) )
29 eleq1 2498 . . . . . . . . . 10  |-  ( x  =  -u ( G `  Z )  ->  (
x  e.  RR  <->  -u ( G `
 Z )  e.  RR ) )
3029anbi2d 686 . . . . . . . . 9  |-  ( x  =  -u ( G `  Z )  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  -u ( G `  Z
)  e.  RR ) ) )
31 stoweidlem23.2 . . . . . . . . . . . . . 14  |-  F/_ t G
32 nfcv 2574 . . . . . . . . . . . . . 14  |-  F/_ t Z
3331, 32nffv 5738 . . . . . . . . . . . . 13  |-  F/_ t
( G `  Z
)
3433nfneg 9307 . . . . . . . . . . . 12  |-  F/_ t -u ( G `  Z
)
3534nfeq2 2585 . . . . . . . . . . 11  |-  F/ t  x  =  -u ( G `  Z )
36 simpl 445 . . . . . . . . . . 11  |-  ( ( x  =  -u ( G `  Z )  /\  t  e.  T
)  ->  x  =  -u ( G `  Z
) )
3735, 36mpteq2da 4297 . . . . . . . . . 10  |-  ( x  =  -u ( G `  Z )  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  -u ( G `  Z )
) )
3837eleq1d 2504 . . . . . . . . 9  |-  ( x  =  -u ( G `  Z )  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  -u ( G `  Z ) )  e.  A ) )
3930, 38imbi12d 313 . . . . . . . 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 3013 . . . . . . 7  |-  ( -u ( G `  Z )  e.  RR  ->  (
( ph  /\  -u ( G `  Z )  e.  RR )  ->  (
t  e.  T  |->  -u ( G `  Z ) )  e.  A ) )
4221, 28, 41sylc 59 . . . . . 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 4301 . . . . . . 7  |-  F/_ t
( t  e.  T  |-> 
-u ( G `  Z ) )
4543, 31, 44stoweidlem8 27747 . . . . . 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 1282 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  ( ( t  e.  T  |-> 
-u ( G `  Z ) ) `  t ) ) )  e.  A )
4727, 46eqeltrrd 2513 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  -u ( G `  Z ) ) )  e.  A
)
4819, 47eqeltrrd 2513 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( G `  Z )
) )  e.  A
)
491, 48syl5eqel 2522 . 2  |-  ( ph  ->  H  e.  A )
50 stoweidlem23.7 . . . . . 6  |-  ( ph  ->  S  e.  T )
5111, 50ffvelrnd 5874 . . . . 5  |-  ( ph  ->  ( G `  S
)  e.  RR )
5251recnd 9119 . . . 4  |-  ( ph  ->  ( G `  S
)  e.  CC )
5315recnd 9119 . . . 4  |-  ( ph  ->  ( G `  Z
)  e.  CC )
54 stoweidlem23.10 . . . 4  |-  ( ph  ->  ( G `  S
)  =/=  ( G `
 Z ) )
5552, 53, 54subne0d 9425 . . 3  |-  ( ph  ->  ( ( G `  S )  -  ( G `  Z )
)  =/=  0 )
5651, 15resubcld 9470 . . . 4  |-  ( ph  ->  ( ( G `  S )  -  ( G `  Z )
)  e.  RR )
57 nfcv 2574 . . . . 5  |-  F/_ t S
5831, 57nffv 5738 . . . . . 6  |-  F/_ t
( G `  S
)
59 nfcv 2574 . . . . . 6  |-  F/_ t  -
6058, 59, 33nfov 6107 . . . . 5  |-  F/_ t
( ( G `  S )  -  ( G `  Z )
)
61 fveq2 5731 . . . . . 6  |-  ( t  =  S  ->  ( G `  t )  =  ( G `  S ) )
6261oveq1d 6099 . . . . 5  |-  ( t  =  S  ->  (
( G `  t
)  -  ( G `
 Z ) )  =  ( ( G `
 S )  -  ( G `  Z ) ) )
6357, 60, 62, 1fvmptf 5824 . . . 4  |-  ( ( S  e.  T  /\  ( ( G `  S )  -  ( G `  Z )
)  e.  RR )  ->  ( H `  S )  =  ( ( G `  S
)  -  ( G `
 Z ) ) )
6450, 56, 63syl2anc 644 . . 3  |-  ( ph  ->  ( H `  S
)  =  ( ( G `  S )  -  ( G `  Z ) ) )
6515, 15resubcld 9470 . . . . 5  |-  ( ph  ->  ( ( G `  Z )  -  ( G `  Z )
)  e.  RR )
6633, 59, 33nfov 6107 . . . . . 6  |-  F/_ t
( ( G `  Z )  -  ( G `  Z )
)
67 fveq2 5731 . . . . . . 7  |-  ( t  =  Z  ->  ( G `  t )  =  ( G `  Z ) )
6867oveq1d 6099 . . . . . 6  |-  ( t  =  Z  ->  (
( G `  t
)  -  ( G `
 Z ) )  =  ( ( G `
 Z )  -  ( G `  Z ) ) )
6932, 66, 68, 1fvmptf 5824 . . . . 5  |-  ( ( Z  e.  T  /\  ( ( G `  Z )  -  ( G `  Z )
)  e.  RR )  ->  ( H `  Z )  =  ( ( G `  Z
)  -  ( G `
 Z ) ) )
7014, 65, 69syl2anc 644 . . . 4  |-  ( ph  ->  ( H `  Z
)  =  ( ( G `  Z )  -  ( G `  Z ) ) )
7153subidd 9404 . . . 4  |-  ( ph  ->  ( ( G `  Z )  -  ( G `  Z )
)  =  0 )
7270, 71eqtrd 2470 . . 3  |-  ( ph  ->  ( H `  Z
)  =  0 )
7355, 64, 723netr4d 2630 . 2  |-  ( ph  ->  ( H `  S
)  =/=  ( H `
 Z ) )
7449, 73, 723jca 1135 1  |-  ( ph  ->  ( H  e.  A  /\  ( H `  S
)  =/=  ( H `
 Z )  /\  ( H `  Z )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   F/wnf 1554    = wceq 1653    e. wcel 1726   F/_wnfc 2561    =/= wne 2601    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995    + caddc 8998    - cmin 9296   -ucneg 9297
This theorem is referenced by:  stoweidlem43  27782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-ltxr 9130  df-sub 9298  df-neg 9299
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