Theorem dprdf1o 18431

Description: Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdf1o.1	|- ( ph -> G dom DProd S )
dprdf1o.2	|- ( ph -> dom S = I )
dprdf1o.3	|- ( ph -> F : J -1-1-onto-> I )

Assertion

Ref	Expression
dprdf1o	|- ( ph -> ( G dom DProd ( S o. F ) /\ ( G DProd ( S o. F ) ) = ( G DProd S ) ) )

Proof of Theorem dprdf1o

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Cntz ` G ) = (Cntz ` G )
2		eqid 2622	. . 3 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2622	. . 3 |- (mrCls ` (SubGrp ` G ) ) = (mrCls ` (SubGrp ` G ) )
4		dprdf1o.1	. . . 4 |- ( ph -> G dom DProd S )
5		dprdgrp 18404	. . . 4 |- ( G dom DProd S -> G e. Grp )
6	4, 5	syl 17	. . 3 |- ( ph -> G e. Grp )
7		dprdf1o.3	. . . . 5 |- ( ph -> F : J -1-1-onto-> I )
8		f1of1 6136	. . . . 5 |- ( F : J -1-1-onto-> I -> F : J -1-1-> I )
9	7, 8	syl 17	. . . 4 |- ( ph -> F : J -1-1-> I )
10		dprdf1o.2	. . . . 5 |- ( ph -> dom S = I )
11	4, 10	dprddomcld 18400	. . . 4 |- ( ph -> I e. _V )
12		f1dmex 7136	. . . 4 |- ( ( F : J -1-1-> I /\ I e. _V ) -> J e. _V )
13	9, 11, 12	syl2anc 693	. . 3 |- ( ph -> J e. _V )
14	4, 10	dprdf2 18406	. . . 4 |- ( ph -> S : I --> (SubGrp ` G ) )
15		f1of 6137	. . . . 5 |- ( F : J -1-1-onto-> I -> F : J --> I )
16	7, 15	syl 17	. . . 4 |- ( ph -> F : J --> I )
17		fco 6058	. . . 4 |- ( ( S : I --> (SubGrp ` G ) /\ F : J --> I ) -> ( S o. F ) : J --> (SubGrp ` G ) )
18	14, 16, 17	syl2anc 693	. . 3 |- ( ph -> ( S o. F ) : J --> (SubGrp ` G ) )
19	4	adantr 481	. . . . 5 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> G dom DProd S )
20	10	adantr 481	. . . . 5 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> dom S = I )
21	16	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> F : J --> I )
22		simpr1 1067	. . . . . 6 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> x e. J )
23	21, 22	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( F ` x ) e. I )
24		simpr2 1068	. . . . . 6 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> y e. J )
25	21, 24	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( F ` y ) e. I )
26		simpr3 1069	. . . . . 6 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> x =/= y )
27	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> F : J -1-1-> I )
28		f1fveq 6519	. . . . . . . 8 |- ( ( F : J -1-1-> I /\ ( x e. J /\ y e. J ) ) -> ( ( F ` x ) = ( F ` y ) <-> x = y ) )
29	27, 22, 24, 28	syl12anc 1324	. . . . . . 7 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( ( F ` x ) = ( F ` y ) <-> x = y ) )
30	29	necon3bid 2838	. . . . . 6 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( ( F ` x ) =/= ( F ` y ) <-> x =/= y ) )
31	26, 30	mpbird 247	. . . . 5 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( F ` x ) =/= ( F ` y ) )
32	19, 20, 23, 25, 31, 1	dprdcntz 18407	. . . 4 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( S ` ( F ` x ) ) C_ ( (Cntz ` G ) ` ( S ` ( F ` y ) ) ) )
33		fvco3 6275	. . . . 5 |- ( ( F : J --> I /\ x e. J ) -> ( ( S o. F ) ` x ) = ( S ` ( F ` x ) ) )
34	21, 22, 33	syl2anc 693	. . . 4 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( ( S o. F ) ` x ) = ( S ` ( F ` x ) ) )
35		fvco3 6275	. . . . . 6 |- ( ( F : J --> I /\ y e. J ) -> ( ( S o. F ) ` y ) = ( S ` ( F ` y ) ) )
36	21, 24, 35	syl2anc 693	. . . . 5 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( ( S o. F ) ` y ) = ( S ` ( F ` y ) ) )
37	36	fveq2d 6195	. . . 4 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( (Cntz ` G ) ` ( ( S o. F ) ` y ) ) = ( (Cntz ` G ) ` ( S ` ( F ` y ) ) ) )
38	32, 34, 37	3sstr4d 3648	. . 3 |- ( ( ph /\ ( x e. J /\ y e. J /\ x =/= y ) ) -> ( ( S o. F ) ` x ) C_ ( (Cntz ` G ) ` ( ( S o. F ) ` y ) ) )
39	16, 33	sylan 488	. . . . . 6 |- ( ( ph /\ x e. J ) -> ( ( S o. F ) ` x ) = ( S ` ( F ` x ) ) )
40		imaco 5640	. . . . . . . . 9 |- ( ( S o. F ) " ( J \ { x } ) ) = ( S " ( F " ( J \ { x } ) ) )
41	7	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. J ) -> F : J -1-1-onto-> I )
42		dff1o3 6143	. . . . . . . . . . . . 13 |- ( F : J -1-1-onto-> I <-> ( F : J -onto-> I /\ Fun `' F ) )
43	42	simprbi 480	. . . . . . . . . . . 12 |- ( F : J -1-1-onto-> I -> Fun `' F )
44		imadif 5973	. . . . . . . . . . . 12 |- ( Fun `' F -> ( F " ( J \ { x } ) ) = ( ( F " J ) \ ( F " { x } ) ) )
45	41, 43, 44	3syl 18	. . . . . . . . . . 11 |- ( ( ph /\ x e. J ) -> ( F " ( J \ { x } ) ) = ( ( F " J ) \ ( F " { x } ) ) )
46		f1ofo 6144	. . . . . . . . . . . . 13 |- ( F : J -1-1-onto-> I -> F : J -onto-> I )
47		foima 6120	. . . . . . . . . . . . 13 |- ( F : J -onto-> I -> ( F " J ) = I )
48	41, 46, 47	3syl 18	. . . . . . . . . . . 12 |- ( ( ph /\ x e. J ) -> ( F " J ) = I )
49		f1ofn 6138	. . . . . . . . . . . . . . 15 |- ( F : J -1-1-onto-> I -> F Fn J )
50	7, 49	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> F Fn J )
51		fnsnfv 6258	. . . . . . . . . . . . . 14 |- ( ( F Fn J /\ x e. J ) -> { ( F ` x ) } = ( F " { x } ) )
52	50, 51	sylan 488	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. J ) -> { ( F ` x ) } = ( F " { x } ) )
53	52	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ph /\ x e. J ) -> ( F " { x } ) = { ( F ` x ) } )
54	48, 53	difeq12d 3729	. . . . . . . . . . 11 |- ( ( ph /\ x e. J ) -> ( ( F " J ) \ ( F " { x } ) ) = ( I \ { ( F ` x ) } ) )
55	45, 54	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ x e. J ) -> ( F " ( J \ { x } ) ) = ( I \ { ( F ` x ) } ) )
56	55	imaeq2d 5466	. . . . . . . . 9 |- ( ( ph /\ x e. J ) -> ( S " ( F " ( J \ { x } ) ) ) = ( S " ( I \ { ( F ` x ) } ) ) )
57	40, 56	syl5eq 2668	. . . . . . . 8 |- ( ( ph /\ x e. J ) -> ( ( S o. F ) " ( J \ { x } ) ) = ( S " ( I \ { ( F ` x ) } ) ) )
58	57	unieqd 4446	. . . . . . 7 |- ( ( ph /\ x e. J ) -> U. ( ( S o. F ) " ( J \ { x } ) ) = U. ( S " ( I \ { ( F ` x ) } ) ) )
59	58	fveq2d 6195	. . . . . 6 |- ( ( ph /\ x e. J ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S o. F ) " ( J \ { x } ) ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { ( F ` x ) } ) ) ) )
60	39, 59	ineq12d 3815	. . . . 5 |- ( ( ph /\ x e. J ) -> ( ( ( S o. F ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S o. F ) " ( J \ { x } ) ) ) ) = ( ( S ` ( F ` x ) ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { ( F ` x ) } ) ) ) ) )
61	4	adantr 481	. . . . . 6 |- ( ( ph /\ x e. J ) -> G dom DProd S )
62	10	adantr 481	. . . . . 6 |- ( ( ph /\ x e. J ) -> dom S = I )
63	16	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. J ) -> ( F ` x ) e. I )
64	61, 62, 63, 2, 3	dprddisj 18408	. . . . 5 |- ( ( ph /\ x e. J ) -> ( ( S ` ( F ` x ) ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { ( F ` x ) } ) ) ) ) = { ( 0g ` G ) } )
65	60, 64	eqtrd 2656	. . . 4 |- ( ( ph /\ x e. J ) -> ( ( ( S o. F ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S o. F ) " ( J \ { x } ) ) ) ) = { ( 0g ` G ) } )
66		eqimss 3657	. . . 4 |- ( ( ( ( S o. F ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S o. F ) " ( J \ { x } ) ) ) ) = { ( 0g ` G ) } -> ( ( ( S o. F ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S o. F ) " ( J \ { x } ) ) ) ) C_ { ( 0g ` G ) } )
67	65, 66	syl 17	. . 3 |- ( ( ph /\ x e. J ) -> ( ( ( S o. F ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S o. F ) " ( J \ { x } ) ) ) ) C_ { ( 0g ` G ) } )
68	1, 2, 3, 6, 13, 18, 38, 67	dmdprdd 18398	. 2 |- ( ph -> G dom DProd ( S o. F ) )
69		rnco2 5642	. . . . . 6 |- ran ( S o. F ) = ( S " ran F )
70		forn 6118	. . . . . . . . 9 |- ( F : J -onto-> I -> ran F = I )
71	7, 46, 70	3syl 18	. . . . . . . 8 |- ( ph -> ran F = I )
72	71	imaeq2d 5466	. . . . . . 7 |- ( ph -> ( S " ran F ) = ( S " I ) )
73		ffn 6045	. . . . . . . 8 |- ( S : I --> (SubGrp ` G ) -> S Fn I )
74		fnima 6010	. . . . . . . 8 |- ( S Fn I -> ( S " I ) = ran S )
75	14, 73, 74	3syl 18	. . . . . . 7 |- ( ph -> ( S " I ) = ran S )
76	72, 75	eqtrd 2656	. . . . . 6 |- ( ph -> ( S " ran F ) = ran S )
77	69, 76	syl5eq 2668	. . . . 5 |- ( ph -> ran ( S o. F ) = ran S )
78	77	unieqd 4446	. . . 4 |- ( ph -> U. ran ( S o. F ) = U. ran S )
79	78	fveq2d 6195	. . 3 |- ( ph -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S o. F ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
80	3	dprdspan 18426	. . . 4 |- ( G dom DProd ( S o. F ) -> ( G DProd ( S o. F ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S o. F ) ) )
81	68, 80	syl 17	. . 3 |- ( ph -> ( G DProd ( S o. F ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S o. F ) ) )
82	3	dprdspan 18426	. . . 4 |- ( G dom DProd S -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
83	4, 82	syl 17	. . 3 |- ( ph -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
84	79, 81, 83	3eqtr4d 2666	. 2 |- ( ph -> ( G DProd ( S o. F ) ) = ( G DProd S ) )
85	68, 84	jca 554	1 |- ( ph -> ( G dom DProd ( S o. F ) /\ ( G DProd ( S o. F ) ) = ( G DProd S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0gc0g 16100  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  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-cmn 18195  df-dprd 18394

This theorem is referenced by:  dprdf1  18432  ablfaclem2  18485