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Theorem stoweidlem8 27624
Description: Lemma for stoweid 27679: 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 959 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  G  e.  A )
2 eleq1 2464 . . . . 5  |-  ( g  =  G  ->  (
g  e.  A  <->  G  e.  A ) )
323anbi3d 1260 . . . 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 2551 . . . . . 6  |-  F/ t  g  =  G
6 fveq1 5686 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  t )  =  ( G `  t ) )
76oveq2d 6056 . . . . . . 7  |-  ( g  =  G  ->  (
( F `  t
)  +  ( g `
 t ) )  =  ( ( F `
 t )  +  ( G `  t
) ) )
87adantr 452 . . . . . 6  |-  ( ( g  =  G  /\  t  e.  T )  ->  ( ( F `  t )  +  ( g `  t ) )  =  ( ( F `  t )  +  ( G `  t ) ) )
95, 8mpteq2da 4254 . . . . 5  |-  ( g  =  G  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) ) )
109eleq1d 2470 . . . 4  |-  ( g  =  G  ->  (
( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  +  ( G `
 t ) ) )  e.  A ) )
113, 10imbi12d 312 . . 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 958 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  F  e.  A )
13 eleq1 2464 . . . . . . 7  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
14133anbi2d 1259 . . . . . 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 2551 . . . . . . . 8  |-  F/ t  f  =  F
17 fveq1 5686 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  t )  =  ( F `  t ) )
1817oveq1d 6055 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f `  t
)  +  ( g `
 t ) )  =  ( ( F `
 t )  +  ( g `  t
) ) )
1918adantr 452 . . . . . . . 8  |-  ( ( f  =  F  /\  t  e.  T )  ->  ( ( f `  t )  +  ( g `  t ) )  =  ( ( F `  t )  +  ( g `  t ) ) )
2016, 19mpteq2da 4254 . . . . . . 7  |-  ( f  =  F  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) ) )
2120eleq1d 2470 . . . . . 6  |-  ( f  =  F  ->  (
( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  e.  A ) )
2214, 21imbi12d 312 . . . . 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 2971 . . . 4  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( g `
 t ) ) )  e.  A ) )
2512, 24mpcom 34 . . 3  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( g `  t ) ) )  e.  A )
2611, 25vtoclg 2971 . 2  |-  ( G  e.  A  ->  (
( ph  /\  F  e.  A  /\  G  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  +  ( G `
 t ) ) )  e.  A ) )
271, 26mpcom 34 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 936    = wceq 1649    e. wcel 1721   F/_wnfc 2527    e. cmpt 4226   ` cfv 5413  (class class class)co 6040    + caddc 8949
This theorem is referenced by:  stoweidlem20  27636  stoweidlem21  27637  stoweidlem22  27638  stoweidlem23  27639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-iota 5377  df-fv 5421  df-ov 6043
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