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Theorem stoweidlem8 29815
Description: Lemma for stoweid 29870: two class variables replace two setvar 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 990 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  G  e.  A )
2 eleq1 2503 . . . . 5  |-  ( g  =  G  ->  (
g  e.  A  <->  G  e.  A ) )
323anbi3d 1295 . . . 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 2605 . . . . . 6  |-  F/ t  g  =  G
6 fveq1 5702 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  t )  =  ( G `  t ) )
76oveq2d 6119 . . . . . . 7  |-  ( g  =  G  ->  (
( F `  t
)  +  ( g `
 t ) )  =  ( ( F `
 t )  +  ( G `  t
) ) )
87adantr 465 . . . . . 6  |-  ( ( g  =  G  /\  t  e.  T )  ->  ( ( F `  t )  +  ( g `  t ) )  =  ( ( F `  t )  +  ( G `  t ) ) )
95, 8mpteq2da 4389 . . . . 5  |-  ( g  =  G  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) ) )
109eleq1d 2509 . . . 4  |-  ( g  =  G  ->  (
( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  +  ( G `
 t ) ) )  e.  A ) )
113, 10imbi12d 320 . . 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 989 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  F  e.  A )
13 eleq1 2503 . . . . . . 7  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
14133anbi2d 1294 . . . . . 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 2605 . . . . . . . 8  |-  F/ t  f  =  F
17 fveq1 5702 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  t )  =  ( F `  t ) )
1817oveq1d 6118 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f `  t
)  +  ( g `
 t ) )  =  ( ( F `
 t )  +  ( g `  t
) ) )
1918adantr 465 . . . . . . . 8  |-  ( ( f  =  F  /\  t  e.  T )  ->  ( ( f `  t )  +  ( g `  t ) )  =  ( ( F `  t )  +  ( g `  t ) ) )
2016, 19mpteq2da 4389 . . . . . . 7  |-  ( f  =  F  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) ) )
2120eleq1d 2509 . . . . . 6  |-  ( f  =  F  ->  (
( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  e.  A ) )
2214, 21imbi12d 320 . . . . 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 3042 . . . 4  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  e.  A ) )
2512, 24mpcom 36 . . 3  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A )
2611, 25vtoclg 3042 . 2  |-  ( G  e.  A  ->  (
( ph  /\  F  e.  A  /\  G  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( G `
 t ) ) )  e.  A ) )
271, 26mpcom 36 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   F/_wnfc 2575    e. cmpt 4362   ` cfv 5430  (class class class)co 6103    + caddc 9297
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-iota 5393  df-fv 5438  df-ov 6106
This theorem is referenced by:  stoweidlem20  29827  stoweidlem21  29828  stoweidlem22  29829  stoweidlem23  29830
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