Theorem dprdval 18402

Description: The value of the internal direct product operation, which is a function mapping the (infinite, but finitely supported) cartesian product of subgroups (which mutually commute and have trivial intersections) to its (group) sum . (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dprdval.0	|- .0. = ( 0g ` G )
dprdval.w	|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }

Assertion

Ref	Expression
dprdval	|- ( ( G dom DProd S /\ dom S = I ) -> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) )

Distinct variable groups:   f, h, i, I   S, f, h, i   f, G, h, i

Allowed substitution hints:   W( f, h, i)   .0. ( f, h, i)

Proof of Theorem dprdval

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

Step	Hyp	Ref	Expression
1		simpl 473	. 2 |- ( ( G dom DProd S /\ dom S = I ) -> G dom DProd S )
2		reldmdprd 18396	. . . . . 6 |- Rel dom DProd
3	2	brrelex2i 5159	. . . . 5 |- ( G dom DProd S -> S e. _V )
4	3	adantr 481	. . . 4 |- ( ( G dom DProd S /\ dom S = I ) -> S e. _V )
5	2	brrelexi 5158	. . . . . 6 |- ( G dom DProd s -> G e. _V )
6		breq1 4656	. . . . . . . 8 |- ( g = G -> ( g dom DProd s <-> G dom DProd s ) )
7		oveq1 6657	. . . . . . . . 9 |- ( g = G -> ( g DProd s ) = ( G DProd s ) )
8		fveq2 6191	. . . . . . . . . . . . . 14 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
9		dprdval.0	. . . . . . . . . . . . . 14 |- .0. = ( 0g ` G )
10	8, 9	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( g = G -> ( 0g ` g ) = .0. )
11	10	breq2d 4665	. . . . . . . . . . . 12 |- ( g = G -> ( h finSupp ( 0g ` g ) <-> h finSupp .0. ) )
12	11	rabbidv 3189	. . . . . . . . . . 11 |- ( g = G -> { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } = { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } )
13		oveq1 6657	. . . . . . . . . . 11 |- ( g = G -> ( g gsumg f ) = ( G gsumg f ) )
14	12, 13	mpteq12dv 4733	. . . . . . . . . 10 |- ( g = G -> ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) = ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) )
15	14	rneqd 5353	. . . . . . . . 9 |- ( g = G -> ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) )
16	7, 15	eqeq12d 2637	. . . . . . . 8 |- ( g = G -> ( ( g DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) <-> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) ) )
17	6, 16	imbi12d 334	. . . . . . 7 |- ( g = G -> ( ( g dom DProd s -> ( g DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) ) <-> ( G dom DProd s -> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) ) ) )
18		df-br 4654	. . . . . . . . 9 |- ( g dom DProd s <-> <. g , s >. e. dom DProd )
19		fvex 6201	. . . . . . . . . . . . . . . . 17 |- ( s ` i ) e. _V
20	19	rgenw 2924	. . . . . . . . . . . . . . . 16 |- A. i e. dom s ( s ` i ) e. _V
21		ixpexg 7932	. . . . . . . . . . . . . . . 16 |- ( A. i e. dom s ( s ` i ) e. _V -> X_ i e. dom s ( s ` i ) e. _V )
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . 15 |- X_ i e. dom s ( s ` i ) e. _V
23	22	mptrabex 6488	. . . . . . . . . . . . . 14 |- ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V
24	23	rnex 7100	. . . . . . . . . . . . 13 |- ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V
25	24	rgen2w 2925	. . . . . . . . . . . 12 |- A. g e. Grp A. s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V
26		df-dprd 18394	. . . . . . . . . . . . 13 |- DProd = ( g e. Grp , s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } |-> ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) )
27	26	fmpt2x 7236	. . . . . . . . . . . 12 |- ( A. g e. Grp A. s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V <-> DProd : U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ) --> _V )
28	25, 27	mpbi 220	. . . . . . . . . . 11 |- DProd : U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ) --> _V
29	28	fdmi 6052	. . . . . . . . . 10 |- dom DProd = U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } )
30	29	eleq2i 2693	. . . . . . . . 9 |- ( <. g , s >. e. dom DProd <-> <. g , s >. e. U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ) )
31		opeliunxp 5170	. . . . . . . . 9 |- ( <. g , s >. e. U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ) <-> ( g e. Grp /\ s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ) )
32	18, 30, 31	3bitri 286	. . . . . . . 8 |- ( g dom DProd s <-> ( g e. Grp /\ s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ) )
33	26	ovmpt4g 6783	. . . . . . . . 9 |- ( ( g e. Grp /\ s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } /\ ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V ) -> ( g DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) )
34	24, 33	mp3an3 1413	. . . . . . . 8 |- ( ( g e. Grp /\ s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. i e. dom h ( A. y e. ( dom h \ { i } ) ( h ` i ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` i ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { i } ) ) ) ) = { ( 0g ` g ) } ) ) } ) -> ( g DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) )
35	32, 34	sylbi 207	. . . . . . 7 |- ( g dom DProd s -> ( g DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) )
36	17, 35	vtoclg 3266	. . . . . 6 |- ( G e. _V -> ( G dom DProd s -> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) ) )
37	5, 36	mpcom 38	. . . . 5 |- ( G dom DProd s -> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) )
38	37	sbcth 3450	. . . 4 |- ( S e. _V -> [. S / s ]. ( G dom DProd s -> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) ) )
39	4, 38	syl 17	. . 3 |- ( ( G dom DProd S /\ dom S = I ) -> [. S / s ]. ( G dom DProd s -> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) ) )
40		simpr 477	. . . . . 6 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> s = S )
41	40	breq2d 4665	. . . . 5 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ( G dom DProd s <-> G dom DProd S ) )
42	40	oveq2d 6666	. . . . . 6 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ( G DProd s ) = ( G DProd S ) )
43	40	dmeqd 5326	. . . . . . . . . . . . 13 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> dom s = dom S )
44		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> dom S = I )
45	43, 44	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> dom s = I )
46	45	ixpeq1d 7920	. . . . . . . . . . 11 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> X_ i e. dom s ( s ` i ) = X_ i e. I ( s ` i ) )
47	40	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ( s ` i ) = ( S ` i ) )
48	47	ixpeq2dv 7924	. . . . . . . . . . 11 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> X_ i e. I ( s ` i ) = X_ i e. I ( S ` i ) )
49	46, 48	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> X_ i e. dom s ( s ` i ) = X_ i e. I ( S ` i ) )
50	49	rabeqdv 3194	. . . . . . . . 9 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } = { h e. X_ i e. I ( S ` i ) | h finSupp .0. } )
51		dprdval.w	. . . . . . . . 9 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
52	50, 51	syl6eqr 2674	. . . . . . . 8 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } = W )
53		eqidd 2623	. . . . . . . 8 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ( G gsumg f ) = ( G gsumg f ) )
54	52, 53	mpteq12dv 4733	. . . . . . 7 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) = ( f e. W |-> ( G gsumg f ) ) )
55	54	rneqd 5353	. . . . . 6 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) = ran ( f e. W |-> ( G gsumg f ) ) )
56	42, 55	eqeq12d 2637	. . . . 5 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ( ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) <-> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) ) )
57	41, 56	imbi12d 334	. . . 4 |- ( ( ( G dom DProd S /\ dom S = I ) /\ s = S ) -> ( ( G dom DProd s -> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) ) <-> ( G dom DProd S -> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) ) ) )
58	4, 57	sbcied 3472	. . 3 |- ( ( G dom DProd S /\ dom S = I ) -> ( [. S / s ]. ( G dom DProd s -> ( G DProd s ) = ran ( f e. { h e. X_ i e. dom s ( s ` i ) | h finSupp .0. } |-> ( G gsumg f ) ) ) <-> ( G dom DProd S -> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) ) ) )
59	39, 58	mpbid 222	. 2 |- ( ( G dom DProd S /\ dom S = I ) -> ( G dom DProd S -> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) ) )
60	1, 59	mpd 15	1 |- ( ( G dom DProd S /\ dom S = I ) -> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   X_cixp 7908   finSupp cfsupp 8275   0gc0g 16100   gsum_g cgsu 16101  mrClscmrc 16243   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

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-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

This theorem is referenced by:  eldprd  18403  dprdlub  18425