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Theorem stoweidlem8 27747
Description: Lemma for stoweid 27802: two class variables replace two set variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem8.1  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem8.2  |-  F/_ t F
stoweidlem8.3  |-  F/_ t G
Assertion
Ref Expression
stoweidlem8  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) )  e.  A )
Distinct variable groups:    f, g,
t    A, f, g    f, F, g    T, f, g    ph, f, g    g, G
Allowed substitution hints:    ph( t)    A( t)    T( t)    F( t)    G( t, f)

Proof of Theorem stoweidlem8
StepHypRef Expression
1 simp3 960 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  G  e.  A )
2 eleq1 2498 . . . . 5  |-  ( g  =  G  ->  (
g  e.  A  <->  G  e.  A ) )
323anbi3d 1261 . . . 4  |-  ( g  =  G  ->  (
( ph  /\  F  e.  A  /\  g  e.  A )  <->  ( ph  /\  F  e.  A  /\  G  e.  A )
) )
4 stoweidlem8.3 . . . . . . 7  |-  F/_ t G
54nfeq2 2585 . . . . . 6  |-  F/ t  g  =  G
6 fveq1 5730 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  t )  =  ( G `  t ) )
76oveq2d 6100 . . . . . . 7  |-  ( g  =  G  ->  (
( F `  t
)  +  ( g `
 t ) )  =  ( ( F `
 t )  +  ( G `  t
) ) )
87adantr 453 . . . . . 6  |-  ( ( g  =  G  /\  t  e.  T )  ->  ( ( F `  t )  +  ( g `  t ) )  =  ( ( F `  t )  +  ( G `  t ) ) )
95, 8mpteq2da 4297 . . . . 5  |-  ( g  =  G  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) ) )
109eleq1d 2504 . . . 4  |-  ( g  =  G  ->  (
( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  +  ( G `
 t ) ) )  e.  A ) )
113, 10imbi12d 313 . . 3  |-  ( g  =  G  ->  (
( ( ph  /\  F  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A
)  <->  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) )  e.  A
) ) )
12 simp2 959 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  F  e.  A )
13 eleq1 2498 . . . . . . 7  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
14133anbi2d 1260 . . . . . 6  |-  ( f  =  F  ->  (
( ph  /\  f  e.  A  /\  g  e.  A )  <->  ( ph  /\  F  e.  A  /\  g  e.  A )
) )
15 stoweidlem8.2 . . . . . . . . 9  |-  F/_ t F
1615nfeq2 2585 . . . . . . . 8  |-  F/ t  f  =  F
17 fveq1 5730 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  t )  =  ( F `  t ) )
1817oveq1d 6099 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f `  t
)  +  ( g `
 t ) )  =  ( ( F `
 t )  +  ( g `  t
) ) )
1918adantr 453 . . . . . . . 8  |-  ( ( f  =  F  /\  t  e.  T )  ->  ( ( f `  t )  +  ( g `  t ) )  =  ( ( F `  t )  +  ( g `  t ) ) )
2016, 19mpteq2da 4297 . . . . . . 7  |-  ( f  =  F  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) ) )
2120eleq1d 2504 . . . . . 6  |-  ( f  =  F  ->  (
( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  e.  A ) )
2214, 21imbi12d 313 . . . . 5  |-  ( f  =  F  ->  (
( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A
)  <->  ( ( ph  /\  F  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A
) ) )
23 stoweidlem8.1 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
2422, 23vtoclg 3013 . . . 4  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  e.  A ) )
2512, 24mpcom 35 . . 3  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A )
2611, 25vtoclg 3013 . 2  |-  ( G  e.  A  ->  (
( ph  /\  F  e.  A  /\  G  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( G `
 t ) ) )  e.  A ) )
271, 26mpcom 35 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    /\ w3a 937    = wceq 1653    e. wcel 1726   F/_wnfc 2561    e. cmpt 4269   ` cfv 5457  (class class class)co 6084    + caddc 8998
This theorem is referenced by:  stoweidlem20  27759  stoweidlem21  27760  stoweidlem22  27761  stoweidlem23  27762
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-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-iota 5421  df-fv 5465  df-ov 6087
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