Theorem dpjfval 18454

Description: Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)

Hypotheses

Ref	Expression
dpjfval.1	|- ( ph -> G dom DProd S )
dpjfval.2	|- ( ph -> dom S = I )
dpjfval.p	|- P = ( GdProj S )
dpjfval.q	|- Q = ( proj1 ` G )

Assertion

Ref	Expression
dpjfval	|- ( ph -> P = ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) )

Distinct variable groups:   i, G   ph, i   i, I   S, i

Allowed substitution hints:   P( i)   Q( i)

Proof of Theorem dpjfval

Dummy variables g s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dpjfval.p	. 2 |- P = ( GdProj S )
2		df-dpj 18395	. . . 4 |- dProj = ( g e. Grp , s e. ( dom DProd " { g } ) |-> ( i e. dom s |-> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) ) )
3	2	a1i 11	. . 3 |- ( ph -> dProj = ( g e. Grp , s e. ( dom DProd " { g } ) |-> ( i e. dom s |-> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) ) ) )
4		simprr 796	. . . . . 6 |- ( ( ph /\ ( g = G /\ s = S ) ) -> s = S )
5	4	dmeqd 5326	. . . . 5 |- ( ( ph /\ ( g = G /\ s = S ) ) -> dom s = dom S )
6		dpjfval.2	. . . . . 6 |- ( ph -> dom S = I )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ ( g = G /\ s = S ) ) -> dom S = I )
8	5, 7	eqtrd 2656	. . . 4 |- ( ( ph /\ ( g = G /\ s = S ) ) -> dom s = I )
9		simprl 794	. . . . . . 7 |- ( ( ph /\ ( g = G /\ s = S ) ) -> g = G )
10	9	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( proj1 ` g ) = ( proj1 ` G ) )
11		dpjfval.q	. . . . . 6 |- Q = ( proj1 ` G )
12	10, 11	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( proj1 ` g ) = Q )
13	4	fveq1d 6193	. . . . 5 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( s ` i ) = ( S ` i ) )
14	8	difeq1d 3727	. . . . . . 7 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( dom s \ { i } ) = ( I \ { i } ) )
15	4, 14	reseq12d 5397	. . . . . 6 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( s |` ( dom s \ { i } ) ) = ( S |` ( I \ { i } ) ) )
16	9, 15	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( g DProd ( s |` ( dom s \ { i } ) ) ) = ( G DProd ( S |` ( I \ { i } ) ) ) )
17	12, 13, 16	oveq123d 6671	. . . 4 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) = ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) )
18	8, 17	mpteq12dv 4733	. . 3 |- ( ( ph /\ ( g = G /\ s = S ) ) -> ( i e. dom s |-> ( ( s ` i ) ( proj1 ` g ) ( g DProd ( s |` ( dom s \ { i } ) ) ) ) ) = ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) )
19		simpr 477	. . . . 5 |- ( ( ph /\ g = G ) -> g = G )
20	19	sneqd 4189	. . . 4 |- ( ( ph /\ g = G ) -> { g } = { G } )
21	20	imaeq2d 5466	. . 3 |- ( ( ph /\ g = G ) -> ( dom DProd " { g } ) = ( dom DProd " { G } ) )
22		dpjfval.1	. . . 4 |- ( ph -> G dom DProd S )
23		dprdgrp 18404	. . . 4 |- ( G dom DProd S -> G e. Grp )
24	22, 23	syl 17	. . 3 |- ( ph -> G e. Grp )
25		reldmdprd 18396	. . . . 5 |- Rel dom DProd
26		elrelimasn 5489	. . . . 5 |- ( Rel dom DProd -> ( S e. ( dom DProd " { G } ) <-> G dom DProd S ) )
27	25, 26	ax-mp 5	. . . 4 |- ( S e. ( dom DProd " { G } ) <-> G dom DProd S )
28	22, 27	sylibr 224	. . 3 |- ( ph -> S e. ( dom DProd " { G } ) )
29	22, 6	dprddomcld 18400	. . . 4 |- ( ph -> I e. _V )
30		mptexg 6484	. . . 4 |- ( I e. _V -> ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) e. _V )
31	29, 30	syl 17	. . 3 |- ( ph -> ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) e. _V )
32	3, 18, 21, 24, 28, 31	ovmpt2dx 6787	. 2 |- ( ph -> ( GdProj S ) = ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) )
33	1, 32	syl5eq 2668	1 |- ( ph -> P = ( i e. I |-> ( ( S ` i ) Q ( G DProd ( S |` ( I \ { i } ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   "cima 5117   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Grpcgrp 17422   proj1cpj1 18050   DProd cdprd 18392  dProjcdpj 18393

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ixp 7909  df-dprd 18394  df-dpj 18395

This theorem is referenced by:  dpjval  18455