Theorem dmdprdsplit2 18445

Description: The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdsplit.2	|- ( ph -> S : I --> (SubGrp ` G ) )
dprdsplit.i	|- ( ph -> ( C i^i D ) = (/) )
dprdsplit.u	|- ( ph -> I = ( C u. D ) )
dmdprdsplit.z	|- Z = (Cntz ` G )
dmdprdsplit.0	|- .0. = ( 0g ` G )
dmdprdsplit2.1	|- ( ph -> G dom DProd ( S |` C ) )
dmdprdsplit2.2	|- ( ph -> G dom DProd ( S |` D ) )
dmdprdsplit2.3	|- ( ph -> ( G DProd ( S |` C ) ) C_ ( Z ` ( G DProd ( S |` D ) ) ) )
dmdprdsplit2.4	|- ( ph -> ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { .0. } )

Assertion

Ref	Expression
dmdprdsplit2	|- ( ph -> G dom DProd S )

Proof of Theorem dmdprdsplit2

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

Step	Hyp	Ref	Expression
1		dmdprdsplit.z	. 2 |- Z = (Cntz ` G )
2		dmdprdsplit.0	. 2 |- .0. = ( 0g ` G )
3		eqid 2622	. 2 |- (mrCls ` (SubGrp ` G ) ) = (mrCls ` (SubGrp ` G ) )
4		dmdprdsplit2.1	. . 3 |- ( ph -> G dom DProd ( S |` C ) )
5		dprdgrp 18404	. . 3 |- ( G dom DProd ( S |` C ) -> G e. Grp )
6	4, 5	syl 17	. 2 |- ( ph -> G e. Grp )
7		dprdsplit.u	. . 3 |- ( ph -> I = ( C u. D ) )
8		dprdsplit.2	. . . . . . 7 |- ( ph -> S : I --> (SubGrp ` G ) )
9		ssun1 3776	. . . . . . . 8 |- C C_ ( C u. D )
10	9, 7	syl5sseqr 3654	. . . . . . 7 |- ( ph -> C C_ I )
11	8, 10	fssresd 6071	. . . . . 6 |- ( ph -> ( S |` C ) : C --> (SubGrp ` G ) )
12		fdm 6051	. . . . . 6 |- ( ( S |` C ) : C --> (SubGrp ` G ) -> dom ( S |` C ) = C )
13	11, 12	syl 17	. . . . 5 |- ( ph -> dom ( S |` C ) = C )
14	4, 13	dprddomcld 18400	. . . 4 |- ( ph -> C e. _V )
15		dmdprdsplit2.2	. . . . 5 |- ( ph -> G dom DProd ( S |` D ) )
16		ssun2 3777	. . . . . . . 8 |- D C_ ( C u. D )
17	16, 7	syl5sseqr 3654	. . . . . . 7 |- ( ph -> D C_ I )
18	8, 17	fssresd 6071	. . . . . 6 |- ( ph -> ( S |` D ) : D --> (SubGrp ` G ) )
19		fdm 6051	. . . . . 6 |- ( ( S |` D ) : D --> (SubGrp ` G ) -> dom ( S |` D ) = D )
20	18, 19	syl 17	. . . . 5 |- ( ph -> dom ( S |` D ) = D )
21	15, 20	dprddomcld 18400	. . . 4 |- ( ph -> D e. _V )
22		unexg 6959	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> ( C u. D ) e. _V )
23	14, 21, 22	syl2anc 693	. . 3 |- ( ph -> ( C u. D ) e. _V )
24	7, 23	eqeltrd 2701	. 2 |- ( ph -> I e. _V )
25	7	eleq2d 2687	. . . . 5 |- ( ph -> ( x e. I <-> x e. ( C u. D ) ) )
26		elun 3753	. . . . 5 |- ( x e. ( C u. D ) <-> ( x e. C \/ x e. D ) )
27	25, 26	syl6bb 276	. . . 4 |- ( ph -> ( x e. I <-> ( x e. C \/ x e. D ) ) )
28		dprdsplit.i	. . . . . . . 8 |- ( ph -> ( C i^i D ) = (/) )
29		dmdprdsplit2.3	. . . . . . . 8 |- ( ph -> ( G DProd ( S |` C ) ) C_ ( Z ` ( G DProd ( S |` D ) ) ) )
30		dmdprdsplit2.4	. . . . . . . 8 |- ( ph -> ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { .0. } )
31	8, 28, 7, 1, 2, 4, 15, 29, 30, 3	dmdprdsplit2lem 18444	. . . . . . 7 |- ( ( ph /\ x e. C ) -> ( ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) )
32		incom 3805	. . . . . . . . 9 |- ( C i^i D ) = ( D i^i C )
33	32, 28	syl5eqr 2670	. . . . . . . 8 |- ( ph -> ( D i^i C ) = (/) )
34		uncom 3757	. . . . . . . . 9 |- ( C u. D ) = ( D u. C )
35	7, 34	syl6eq 2672	. . . . . . . 8 |- ( ph -> I = ( D u. C ) )
36		dprdsubg 18423	. . . . . . . . . 10 |- ( G dom DProd ( S |` C ) -> ( G DProd ( S |` C ) ) e. (SubGrp ` G ) )
37	4, 36	syl 17	. . . . . . . . 9 |- ( ph -> ( G DProd ( S |` C ) ) e. (SubGrp ` G ) )
38		dprdsubg 18423	. . . . . . . . . 10 |- ( G dom DProd ( S |` D ) -> ( G DProd ( S |` D ) ) e. (SubGrp ` G ) )
39	15, 38	syl 17	. . . . . . . . 9 |- ( ph -> ( G DProd ( S |` D ) ) e. (SubGrp ` G ) )
40	1, 37, 39, 29	cntzrecd 18091	. . . . . . . 8 |- ( ph -> ( G DProd ( S |` D ) ) C_ ( Z ` ( G DProd ( S |` C ) ) ) )
41		incom 3805	. . . . . . . . 9 |- ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = ( ( G DProd ( S |` D ) ) i^i ( G DProd ( S |` C ) ) )
42	41, 30	syl5eqr 2670	. . . . . . . 8 |- ( ph -> ( ( G DProd ( S |` D ) ) i^i ( G DProd ( S |` C ) ) ) = { .0. } )
43	8, 33, 35, 1, 2, 15, 4, 40, 42, 3	dmdprdsplit2lem 18444	. . . . . . 7 |- ( ( ph /\ x e. D ) -> ( ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) )
44	31, 43	jaodan 826	. . . . . 6 |- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) )
45	44	simpld 475	. . . . 5 |- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) )
46	45	ex 450	. . . 4 |- ( ph -> ( ( x e. C \/ x e. D ) -> ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) ) )
47	27, 46	sylbid 230	. . 3 |- ( ph -> ( x e. I -> ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) ) )
48	47	3imp2 1282	. 2 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( S ` x ) C_ ( Z ` ( S ` y ) ) )
49	27	biimpa 501	. . 3 |- ( ( ph /\ x e. I ) -> ( x e. C \/ x e. D ) )
50	31	simprd 479	. . . 4 |- ( ( ph /\ x e. C ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } )
51	43	simprd 479	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } )
52	50, 51	jaodan 826	. . 3 |- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } )
53	49, 52	syldan 487	. 2 |- ( ( ph /\ x e. I ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } )
54	1, 2, 3, 6, 24, 8, 48, 53	dmdprdd 18398	1 |- ( ph -> G dom DProd S )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   "cima 5117   -->wf 5884   ` 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-lsm 18051  df-cmn 18195  df-dprd 18394

This theorem is referenced by:  dmdprdsplit  18446  pgpfaclem1  18480