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Theorem stoweidlem22 27749
Description: If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem22.8  |-  F/ t
ph
stoweidlem22.9  |-  F/_ t F
stoweidlem22.10  |-  F/_ t G
stoweidlem22.1  |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t )
) )
stoweidlem22.2  |-  I  =  ( t  e.  T  |-> 
-u 1 )
stoweidlem22.3  |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
stoweidlem22.4  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem22.5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem22.6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem22.7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
Assertion
Ref Expression
stoweidlem22  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Distinct variable groups:    f, g,
t, A    f, F, g    f, G, g    f, I, g    T, f, g, t    ph, f, g    g, L    x, t, A    x, T    ph, x
Allowed substitution hints:    ph( t)    F( x, t)    G( x, t)    H( x, t, f, g)    I( x, t)    L( x, t, f)

Proof of Theorem stoweidlem22
StepHypRef Expression
1 stoweidlem22.8 . . . 4  |-  F/ t
ph
2 stoweidlem22.9 . . . . 5  |-  F/_ t F
32nfel1 2584 . . . 4  |-  F/ t  F  e.  A
4 stoweidlem22.10 . . . . 5  |-  F/_ t G
54nfel1 2584 . . . 4  |-  F/ t  G  e.  A
61, 3, 5nf3an 1850 . . 3  |-  F/ t ( ph  /\  F  e.  A  /\  G  e.  A )
7 simpr 449 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  t  e.  T )
8 simpl1 961 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ph )
9 stoweidlem22.2 . . . . . . . . . . . 12  |-  I  =  ( t  e.  T  |-> 
-u 1 )
10 1re 9092 . . . . . . . . . . . . . 14  |-  1  e.  RR
1110renegcli 9364 . . . . . . . . . . . . 13  |-  -u 1  e.  RR
12 stoweidlem22.7 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
1312stoweidlem4 27731 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -u 1  e.  RR )  ->  (
t  e.  T  |->  -u
1 )  e.  A
)
1411, 13mpan2 654 . . . . . . . . . . . 12  |-  ( ph  ->  ( t  e.  T  |-> 
-u 1 )  e.  A )
159, 14syl5eqel 2522 . . . . . . . . . . 11  |-  ( ph  ->  I  e.  A )
16 eleq1 2498 . . . . . . . . . . . . . . 15  |-  ( f  =  I  ->  (
f  e.  A  <->  I  e.  A ) )
1716anbi2d 686 . . . . . . . . . . . . . 14  |-  ( f  =  I  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  I  e.  A ) ) )
18 feq1 5578 . . . . . . . . . . . . . 14  |-  ( f  =  I  ->  (
f : T --> RR  <->  I : T
--> RR ) )
1917, 18imbi12d 313 . . . . . . . . . . . . 13  |-  ( f  =  I  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  I  e.  A )  ->  I : T --> RR ) ) )
20 stoweidlem22.4 . . . . . . . . . . . . 13  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
2119, 20vtoclg 3013 . . . . . . . . . . . 12  |-  ( I  e.  A  ->  (
( ph  /\  I  e.  A )  ->  I : T --> RR ) )
2221anabsi7 794 . . . . . . . . . . 11  |-  ( (
ph  /\  I  e.  A )  ->  I : T --> RR )
2315, 22mpdan 651 . . . . . . . . . 10  |-  ( ph  ->  I : T --> RR )
248, 23syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  I : T --> RR )
2524, 7ffvelrnd 5873 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
I `  t )  e.  RR )
26 simpl3 963 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  G  e.  A )
27 eleq1 2498 . . . . . . . . . . . . . . 15  |-  ( f  =  G  ->  (
f  e.  A  <->  G  e.  A ) )
2827anbi2d 686 . . . . . . . . . . . . . 14  |-  ( f  =  G  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  G  e.  A ) ) )
29 feq1 5578 . . . . . . . . . . . . . 14  |-  ( f  =  G  ->  (
f : T --> RR  <->  G : T
--> RR ) )
3028, 29imbi12d 313 . . . . . . . . . . . . 13  |-  ( f  =  G  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  G  e.  A )  ->  G : T --> RR ) ) )
3130, 20vtoclg 3013 . . . . . . . . . . . 12  |-  ( G  e.  A  ->  (
( ph  /\  G  e.  A )  ->  G : T --> RR ) )
3231anabsi7 794 . . . . . . . . . . 11  |-  ( (
ph  /\  G  e.  A )  ->  G : T --> RR )
33323adant3 978 . . . . . . . . . 10  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  G : T
--> RR )
34 simp3 960 . . . . . . . . . 10  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  t  e.  T )
3533, 34ffvelrnd 5873 . . . . . . . . 9  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  ( G `  t )  e.  RR )
368, 26, 7, 35syl3anc 1185 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( G `  t )  e.  RR )
3725, 36remulcld 9118 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( I `  t
)  x.  ( G `
 t ) )  e.  RR )
38 stoweidlem22.3 . . . . . . . 8  |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
3938fvmpt2 5814 . . . . . . 7  |-  ( ( t  e.  T  /\  ( ( I `  t )  x.  ( G `  t )
)  e.  RR )  ->  ( L `  t )  =  ( ( I `  t
)  x.  ( G `
 t ) ) )
407, 37, 39syl2anc 644 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( L `  t )  =  ( ( I `
 t )  x.  ( G `  t
) ) )
419fvmpt2 5814 . . . . . . . . 9  |-  ( ( t  e.  T  /\  -u 1  e.  RR )  ->  ( I `  t )  =  -u
1 )
4211, 41mpan2 654 . . . . . . . 8  |-  ( t  e.  T  ->  (
I `  t )  =  -u 1 )
4342adantl 454 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
I `  t )  =  -u 1 )
4443oveq1d 6098 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( I `  t
)  x.  ( G `
 t ) )  =  ( -u 1  x.  ( G `  t
) ) )
4536recnd 9116 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( G `  t )  e.  CC )
4645mulm1d 9487 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( -u 1  x.  ( G `
 t ) )  =  -u ( G `  t ) )
4740, 44, 463eqtrd 2474 . . . . 5  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( L `  t )  =  -u ( G `  t ) )
4847oveq2d 6099 . . . 4  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  +  ( L `
 t ) )  =  ( ( F `
 t )  + 
-u ( G `  t ) ) )
49 simpl2 962 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  F  e.  A )
50 eleq1 2498 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
5150anbi2d 686 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  F  e.  A ) ) )
52 feq1 5578 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f : T --> RR  <->  F : T
--> RR ) )
5351, 52imbi12d 313 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  F  e.  A )  ->  F : T --> RR ) ) )
5453, 20vtoclg 3013 . . . . . . . . 9  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A )  ->  F : T --> RR ) )
5554anabsi7 794 . . . . . . . 8  |-  ( (
ph  /\  F  e.  A )  ->  F : T --> RR )
568, 49, 55syl2anc 644 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  F : T --> RR )
5756, 7ffvelrnd 5873 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
5857recnd 9116 . . . . 5  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
5958, 45negsubd 9419 . . . 4  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  +  -u ( G `  t )
)  =  ( ( F `  t )  -  ( G `  t ) ) )
6048, 59eqtr2d 2471 . . 3  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  -  ( G `
 t ) )  =  ( ( F `
 t )  +  ( L `  t
) ) )
616, 60mpteq2da 4296 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  +  ( L `  t
) ) ) )
62153ad2ant1 979 . . . . 5  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  I  e.  A )
63 nfmpt1 4300 . . . . . . . 8  |-  F/_ t
( t  e.  T  |-> 
-u 1 )
649, 63nfcxfr 2571 . . . . . . 7  |-  F/_ t
I
6564nfeq2 2585 . . . . . 6  |-  F/ t  f  =  I
664nfeq2 2585 . . . . . 6  |-  F/ t  g  =  G
67 stoweidlem22.6 . . . . . 6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
6865, 66, 67stoweidlem6 27733 . . . . 5  |-  ( (
ph  /\  I  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t ) ) )  e.  A )
6962, 68syld3an2 1232 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t ) ) )  e.  A )
7038, 69syl5eqel 2522 . . 3  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  L  e.  A )
71 stoweidlem22.5 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
72 nfmpt1 4300 . . . . 5  |-  F/_ t
( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
7338, 72nfcxfr 2571 . . . 4  |-  F/_ t L
7471, 2, 73stoweidlem8 27735 . . 3  |-  ( (
ph  /\  F  e.  A  /\  L  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( L `  t ) ) )  e.  A )
7570, 74syld3an3 1230 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( L `  t ) ) )  e.  A )
7661, 75eqeltrd 2512 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
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    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083   RRcr 8991   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294
This theorem is referenced by:  stoweidlem33  27760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-ltxr 9127  df-sub 9295  df-neg 9296
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