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Theorem stoweidlem6 29801
Description: Lemma for stoweid 29858: two class variables replace two setvar 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 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 stoweidlem6.2 . . . . . 6  |-  F/ t  g  =  G
5 fveq1 5690 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  t )  =  ( G `  t ) )
65oveq2d 6107 . . . . . . 7  |-  ( g  =  G  ->  (
( F `  t
)  x.  ( g `
 t ) )  =  ( ( F `
 t )  x.  ( G `  t
) ) )
76adantr 465 . . . . . 6  |-  ( ( g  =  G  /\  t  e.  T )  ->  ( ( F `  t )  x.  (
g `  t )
)  =  ( ( F `  t )  x.  ( G `  t ) ) )
84, 7mpteq2da 4377 . . . . 5  |-  ( g  =  G  ->  (
t  e.  T  |->  ( ( F `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  x.  ( G `  t ) ) ) )
98eleq1d 2509 . . . 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 320 . . 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 989 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  F  e.  A )
12 eleq1 2503 . . . . . . 7  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
13123anbi2d 1294 . . . . . 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 5690 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  t )  =  ( F `  t ) )
1615oveq1d 6106 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f `  t
)  x.  ( g `
 t ) )  =  ( ( F `
 t )  x.  ( g `  t
) ) )
1716adantr 465 . . . . . . . 8  |-  ( ( f  =  F  /\  t  e.  T )  ->  ( ( f `  t )  x.  (
g `  t )
)  =  ( ( F `  t )  x.  ( g `  t ) ) )
1814, 17mpteq2da 4377 . . . . . . 7  |-  ( f  =  F  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( F `  t )  x.  ( g `  t ) ) ) )
1918eleq1d 2509 . . . . . 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 320 . . . . 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 3030 . . . 4  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  x.  ( g `
 t ) ) )  e.  A ) )
2311, 22mpcom 36 . . 3  |-  ( (
ph  /\  F  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( g `  t ) ) )  e.  A )
2410, 23vtoclg 3030 . 2  |-  ( G  e.  A  ->  (
( ph  /\  F  e.  A  /\  G  e.  A )  ->  (
t  e.  T  |->  ( ( F `  t
)  x.  ( G `
 t ) ) )  e.  A ) )
251, 24mpcom 36 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( G `  t ) ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369   F/wnf 1589    e. wcel 1756    e. cmpt 4350   ` cfv 5418  (class class class)co 6091    x. cmul 9287
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 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-iota 5381  df-fv 5426  df-ov 6094
This theorem is referenced by:  stoweidlem19  29814  stoweidlem22  29817  stoweidlem32  29827  stoweidlem36  29831
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