Theorem stoweidlem6 40223

Description: Lemma for stoweid 40280: 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

Step	Hyp	Ref	Expression
1		simp3 1063	. 2 |- ( ( ph /\ F e. A /\ G e. A ) -> G e. A )
2		eleq1 2689	. . . . 5 |- ( g = G -> ( g e. A <-> G e. A ) )
3	2	3anbi3d 1405	. . . 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 6190	. . . . . . . 8 |- ( g = G -> ( g ` t ) = ( G ` t ) )
6	5	oveq2d 6666	. . . . . . 7 |- ( g = G -> ( ( F ` t ) x. ( g ` t ) ) = ( ( F ` t ) x. ( G ` t ) ) )
7	6	adantr 481	. . . . . 6 |- ( ( g = G /\ t e. T ) -> ( ( F ` t ) x. ( g ` t ) ) = ( ( F ` t ) x. ( G ` t ) ) )
8	4, 7	mpteq2da 4743	. . . . 5 |- ( g = G -> ( t e. T |-> ( ( F ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( F ` t ) x. ( G ` t ) ) ) )
9	8	eleq1d 2686	. . . 4 |- ( g = G -> ( ( t e. T |-> ( ( F ` t ) x. ( g ` t ) ) ) e. A <-> ( t e. T |-> ( ( F ` t ) x. ( G ` t ) ) ) e. A ) )
10	3, 9	imbi12d 334	. . 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 1062	. . . 4 |- ( ( ph /\ F e. A /\ g e. A ) -> F e. A )
12		eleq1 2689	. . . . . . 7 |- ( f = F -> ( f e. A <-> F e. A ) )
13	12	3anbi2d 1404	. . . . . 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 6190	. . . . . . . . . 10 |- ( f = F -> ( f ` t ) = ( F ` t ) )
16	15	oveq1d 6665	. . . . . . . . 9 |- ( f = F -> ( ( f ` t ) x. ( g ` t ) ) = ( ( F ` t ) x. ( g ` t ) ) )
17	16	adantr 481	. . . . . . . 8 |- ( ( f = F /\ t e. T ) -> ( ( f ` t ) x. ( g ` t ) ) = ( ( F ` t ) x. ( g ` t ) ) )
18	14, 17	mpteq2da 4743	. . . . . . 7 |- ( f = F -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( F ` t ) x. ( g ` t ) ) ) )
19	18	eleq1d 2686	. . . . . 6 |- ( f = F -> ( ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A <-> ( t e. T |-> ( ( F ` t ) x. ( g ` t ) ) ) e. A ) )
20	13, 19	imbi12d 334	. . . . 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 )
22	20, 21	vtoclg 3266	. . . 4 |- ( F e. A -> ( ( ph /\ F e. A /\ g e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( g ` t ) ) ) e. A ) )
23	11, 22	mpcom 38	. . 3 |- ( ( ph /\ F e. A /\ g e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( g ` t ) ) ) e. A )
24	10, 23	vtoclg 3266	. 2 |- ( G e. A -> ( ( ph /\ F e. A /\ G e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( G ` t ) ) ) e. A ) )
25	1, 24	mpcom 38	1 |- ( ( ph /\ F e. A /\ G e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( G ` t ) ) ) e. A )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  stoweidlem19  40236  stoweidlem22  40239  stoweidlem32  40249  stoweidlem36  40253