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Theorem stoweidlem23 31351
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 2539 . . . . . . . . . . . 12  |-  ( f  =  G  ->  (
f  e.  A  <->  G  e.  A ) )
65anbi2d 703 . . . . . . . . . . 11  |-  ( f  =  G  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  G  e.  A ) ) )
7 feq1 5713 . . . . . . . . . . 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 3171 . . . . . . . . 9  |-  ( G  e.  A  ->  (
( ph  /\  G  e.  A )  ->  G : T --> RR ) )
113, 4, 10sylc 60 . . . . . . . 8  |-  ( ph  ->  G : T --> RR )
1211fnvinran 30995 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  e.  RR )
1312recnd 9622 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  e.  CC )
14 stoweidlem23.8 . . . . . . . . 9  |-  ( ph  ->  Z  e.  T )
1511, 14ffvelrnd 6022 . . . . . . . 8  |-  ( ph  ->  ( G `  Z
)  e.  RR )
1615adantr 465 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  Z )  e.  RR )
1716recnd 9622 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  Z )  e.  CC )
1813, 17negsubd 9936 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  +  -u ( G `  Z )
)  =  ( ( G `  t )  -  ( G `  Z ) ) )
192, 18mpteq2da 4532 . . . 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 9986 . . . . . . . . 9  |-  ( ph  -> 
-u ( G `  Z )  e.  RR )
2221adantr 465 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  -u ( G `  Z )  e.  RR )
23 eqid 2467 . . . . . . . . 9  |-  ( t  e.  T  |->  -u ( G `  Z )
)  =  ( t  e.  T  |->  -u ( G `  Z )
)
2423fvmpt2 5957 . . . . . . . 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 6300 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  +  ( ( t  e.  T  |->  -u ( G `  Z ) ) `  t ) )  =  ( ( G `  t )  +  -u ( G `  Z ) ) )
272, 26mpteq2da 4532 . . . . 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 2539 . . . . . . . . . 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 2629 . . . . . . . . . . . . . 14  |-  F/_ t Z
3331, 32nffv 5873 . . . . . . . . . . . . 13  |-  F/_ t
( G `  Z
)
3433nfneg 9816 . . . . . . . . . . . 12  |-  F/_ t -u ( G `  Z
)
3534nfeq2 2646 . . . . . . . . . . 11  |-  F/ t  x  =  -u ( G `  Z )
36 simpl 457 . . . . . . . . . . 11  |-  ( ( x  =  -u ( G `  Z )  /\  t  e.  T
)  ->  x  =  -u ( G `  Z
) )
3735, 36mpteq2da 4532 . . . . . . . . . 10  |-  ( x  =  -u ( G `  Z )  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  -u ( G `  Z )
) )
3837eleq1d 2536 . . . . . . . . 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 3171 . . . . . . 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 4536 . . . . . . 7  |-  F/_ t
( t  e.  T  |-> 
-u ( G `  Z ) )
4543, 31, 44stoweidlem8 31336 . . . . . 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 1326 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  ( ( t  e.  T  |-> 
-u ( G `  Z ) ) `  t ) ) )  e.  A )
4727, 46eqeltrrd 2556 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  +  -u ( G `  Z ) ) )  e.  A
)
4819, 47eqeltrrd 2556 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( G `  Z )
) )  e.  A
)
491, 48syl5eqel 2559 . 2  |-  ( ph  ->  H  e.  A )
50 stoweidlem23.7 . . . . . 6  |-  ( ph  ->  S  e.  T )
5111, 50ffvelrnd 6022 . . . . 5  |-  ( ph  ->  ( G `  S
)  e.  RR )
5251recnd 9622 . . . 4  |-  ( ph  ->  ( G `  S
)  e.  CC )
5315recnd 9622 . . . 4  |-  ( ph  ->  ( G `  Z
)  e.  CC )
54 stoweidlem23.10 . . . 4  |-  ( ph  ->  ( G `  S
)  =/=  ( G `
 Z ) )
5552, 53, 54subne0d 9939 . . 3  |-  ( ph  ->  ( ( G `  S )  -  ( G `  Z )
)  =/=  0 )
5651, 15resubcld 9987 . . . 4  |-  ( ph  ->  ( ( G `  S )  -  ( G `  Z )
)  e.  RR )
57 nfcv 2629 . . . . 5  |-  F/_ t S
5831, 57nffv 5873 . . . . . 6  |-  F/_ t
( G `  S
)
59 nfcv 2629 . . . . . 6  |-  F/_ t  -
6058, 59, 33nfov 6307 . . . . 5  |-  F/_ t
( ( G `  S )  -  ( G `  Z )
)
61 fveq2 5866 . . . . . 6  |-  ( t  =  S  ->  ( G `  t )  =  ( G `  S ) )
6261oveq1d 6299 . . . . 5  |-  ( t  =  S  ->  (
( G `  t
)  -  ( G `
 Z ) )  =  ( ( G `
 S )  -  ( G `  Z ) ) )
6357, 60, 62, 1fvmptf 5966 . . . 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 9987 . . . . 5  |-  ( ph  ->  ( ( G `  Z )  -  ( G `  Z )
)  e.  RR )
6633, 59, 33nfov 6307 . . . . . 6  |-  F/_ t
( ( G `  Z )  -  ( G `  Z )
)
67 fveq2 5866 . . . . . . 7  |-  ( t  =  Z  ->  ( G `  t )  =  ( G `  Z ) )
6867oveq1d 6299 . . . . . 6  |-  ( t  =  Z  ->  (
( G `  t
)  -  ( G `
 Z ) )  =  ( ( G `
 Z )  -  ( G `  Z ) ) )
6932, 66, 68, 1fvmptf 5966 . . . . 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 9918 . . . 4  |-  ( ph  ->  ( ( G `  Z )  -  ( G `  Z )
)  =  0 )
7270, 71eqtrd 2508 . . 3  |-  ( ph  ->  ( H `  Z
)  =  0 )
7355, 64, 723netr4d 2772 . 2  |-  ( ph  ->  ( H `  S
)  =/=  ( H `
 Z ) )
7449, 73, 723jca 1176 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 973    = wceq 1379   F/wnf 1599    e. wcel 1767   F/_wnfc 2615    =/= wne 2662    |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492    + caddc 9495    - cmin 9805   -ucneg 9806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-sub 9807  df-neg 9808
This theorem is referenced by:  stoweidlem43  31371
  Copyright terms: Public domain W3C validator