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Theorem stoweidlem6 27733
Description: Lemma for stoweid 27790: two class variables replace two set variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem6.1  |-  F/ t  f  =  F
stoweidlem6.2  |-  F/ t  g  =  G
stoweidlem6.3  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
Assertion
Ref Expression
stoweidlem6  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( 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 stoweidlem6
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 stoweidlem6.2 . . . . . 6  |-  F/ t  g  =  G
5 fveq1 5729 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  t )  =  ( G `  t ) )
65oveq2d 6099 . . . . . . 7  |-  ( g  =  G  ->  (
( F `  t
)  x.  ( g `
 t ) )  =  ( ( F `
 t )  x.  ( G `  t
) ) )
76adantr 453 . . . . . 6  |-  ( ( g  =  G  /\  t  e.  T )  ->  ( ( F `  t )  x.  (
g `  t )
)  =  ( ( F `  t )  x.  ( G `  t ) ) )
84, 7mpteq2da 4296 . . . . 5  |-  ( g  =  G  ->  (
t  e.  T  |->  ( ( F `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  x.  ( G `  t ) ) ) )
98eleq1d 2504 . . . 4  |-  ( g  =  G  ->  (
( t  e.  T  |->  ( ( F `  t )  x.  (
g `  t )
) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  x.  ( G `
 t ) ) )  e.  A ) )
103, 9imbi12d 313 . . 3  |-  ( g  =  G  ->  (
( ( ph  /\  F  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  x.  (
g `  t )
) )  e.  A
)  <->  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( G `  t )
) )  e.  A
) ) )
11 simp2 959 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  F  e.  A )
12 eleq1 2498 . . . . . . 7  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
13123anbi2d 1260 . . . . . 6  |-  ( f  =  F  ->  (
( ph  /\  f  e.  A  /\  g  e.  A )  <->  ( ph  /\  F  e.  A  /\  g  e.  A )
) )
14 stoweidlem6.1 . . . . . . . 8  |-  F/ t  f  =  F
15 fveq1 5729 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  t )  =  ( F `  t ) )
1615oveq1d 6098 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f `  t
)  x.  ( g `
 t ) )  =  ( ( F `
 t )  x.  ( g `  t
) ) )
1716adantr 453 . . . . . . . 8  |-  ( ( f  =  F  /\  t  e.  T )  ->  ( ( f `  t )  x.  (
g `  t )
)  =  ( ( F `  t )  x.  ( g `  t ) ) )
1814, 17mpteq2da 4296 . . . . . . 7  |-  ( f  =  F  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  x.  ( g `  t ) ) ) )
1918eleq1d 2504 . . . . . 6  |-  ( f  =  F  ->  (
( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A  <->  ( t  e.  T  |->  ( ( F `  t
)  x.  ( g `
 t ) ) )  e.  A ) )
2013, 19imbi12d 313 . . . . 5  |-  ( f  =  F  ->  (
( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A
)  <->  ( ( ph  /\  F  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  x.  (
g `  t )
) )  e.  A
) ) )
21 stoweidlem6.3 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
2220, 21vtoclg 3013 . . . 4  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  x.  ( g `
 t ) ) )  e.  A ) )
2311, 22mpcom 35 . . 3  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( g `  t ) ) )  e.  A )
2410, 23vtoclg 3013 . 2  |-  ( G  e.  A  ->  (
( ph  /\  F  e.  A  /\  G  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  x.  ( G `
 t ) ) )  e.  A ) )
251, 24mpcom 35 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( G `  t ) ) )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937   F/wnf 1554    = wceq 1653    e. wcel 1726    e. cmpt 4268   ` cfv 5456  (class class class)co 6083    x. cmul 8997
This theorem is referenced by:  stoweidlem19  27746  stoweidlem22  27749  stoweidlem32  27759  stoweidlem36  27763
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-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 4215  df-opab 4269  df-mpt 4270  df-iota 5420  df-fv 5464  df-ov 6086
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