Theorem dprdsplit 18447

Description: The direct product is the binary subgroup product ("sum") of the direct products of the partition. (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 ) )
dprdsplit.s	|- .(+) = ( LSSum ` G )
dprdsplit.1	|- ( ph -> G dom DProd S )

Assertion

Ref	Expression
dprdsplit	|- ( ph -> ( G DProd S ) = ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )

Proof of Theorem dprdsplit

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dprdsplit.1	. . 3 |- ( ph -> G dom DProd S )
2		dprdsplit.2	. . . 4 |- ( ph -> S : I --> (SubGrp ` G ) )
3		fdm 6051	. . . 4 |- ( S : I --> (SubGrp ` G ) -> dom S = I )
4	2, 3	syl 17	. . 3 |- ( ph -> dom S = I )
5		ssun1 3776	. . . . . . . 8 |- C C_ ( C u. D )
6		dprdsplit.u	. . . . . . . 8 |- ( ph -> I = ( C u. D ) )
7	5, 6	syl5sseqr 3654	. . . . . . 7 |- ( ph -> C C_ I )
8	1, 4, 7	dprdres 18427	. . . . . 6 |- ( ph -> ( G dom DProd ( S |` C ) /\ ( G DProd ( S |` C ) ) C_ ( G DProd S ) ) )
9	8	simpld 475	. . . . 5 |- ( ph -> G dom DProd ( S |` C ) )
10		dprdsubg 18423	. . . . 5 |- ( G dom DProd ( S |` C ) -> ( G DProd ( S |` C ) ) e. (SubGrp ` G ) )
11	9, 10	syl 17	. . . 4 |- ( ph -> ( G DProd ( S |` C ) ) e. (SubGrp ` G ) )
12		ssun2 3777	. . . . . . . 8 |- D C_ ( C u. D )
13	12, 6	syl5sseqr 3654	. . . . . . 7 |- ( ph -> D C_ I )
14	1, 4, 13	dprdres 18427	. . . . . 6 |- ( ph -> ( G dom DProd ( S |` D ) /\ ( G DProd ( S |` D ) ) C_ ( G DProd S ) ) )
15	14	simpld 475	. . . . 5 |- ( ph -> G dom DProd ( S |` D ) )
16		dprdsubg 18423	. . . . 5 |- ( G dom DProd ( S |` D ) -> ( G DProd ( S |` D ) ) e. (SubGrp ` G ) )
17	15, 16	syl 17	. . . 4 |- ( ph -> ( G DProd ( S |` D ) ) e. (SubGrp ` G ) )
18		dprdsplit.i	. . . . . . 7 |- ( ph -> ( C i^i D ) = (/) )
19		eqid 2622	. . . . . . 7 |- (Cntz ` G ) = (Cntz ` G )
20		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
21	2, 18, 6, 19, 20	dmdprdsplit 18446	. . . . . 6 |- ( ph -> ( G dom DProd S <-> ( ( G dom DProd ( S |` C ) /\ G dom DProd ( S |` D ) ) /\ ( G DProd ( S |` C ) ) C_ ( (Cntz ` G ) ` ( G DProd ( S |` D ) ) ) /\ ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { ( 0g ` G ) } ) ) )
22	1, 21	mpbid 222	. . . . 5 |- ( ph -> ( ( G dom DProd ( S |` C ) /\ G dom DProd ( S |` D ) ) /\ ( G DProd ( S |` C ) ) C_ ( (Cntz ` G ) ` ( G DProd ( S |` D ) ) ) /\ ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { ( 0g ` G ) } ) )
23	22	simp2d 1074	. . . 4 |- ( ph -> ( G DProd ( S |` C ) ) C_ ( (Cntz ` G ) ` ( G DProd ( S |` D ) ) ) )
24		dprdsplit.s	. . . . 5 |- .(+) = ( LSSum ` G )
25	24, 19	lsmsubg 18069	. . . 4 |- ( ( ( G DProd ( S |` C ) ) e. (SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. (SubGrp ` G ) /\ ( G DProd ( S |` C ) ) C_ ( (Cntz ` G ) ` ( G DProd ( S |` D ) ) ) ) -> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) e. (SubGrp ` G ) )
26	11, 17, 23, 25	syl3anc 1326	. . 3 |- ( ph -> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) e. (SubGrp ` G ) )
27	6	eleq2d 2687	. . . . . 6 |- ( ph -> ( x e. I <-> x e. ( C u. D ) ) )
28		elun 3753	. . . . . 6 |- ( x e. ( C u. D ) <-> ( x e. C \/ x e. D ) )
29	27, 28	syl6bb 276	. . . . 5 |- ( ph -> ( x e. I <-> ( x e. C \/ x e. D ) ) )
30	29	biimpa 501	. . . 4 |- ( ( ph /\ x e. I ) -> ( x e. C \/ x e. D ) )
31		fvres 6207	. . . . . . . 8 |- ( x e. C -> ( ( S |` C ) ` x ) = ( S ` x ) )
32	31	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. C ) -> ( ( S |` C ) ` x ) = ( S ` x ) )
33	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. C ) -> G dom DProd ( S |` C ) )
34	2, 7	fssresd 6071	. . . . . . . . . 10 |- ( ph -> ( S |` C ) : C --> (SubGrp ` G ) )
35		fdm 6051	. . . . . . . . . 10 |- ( ( S |` C ) : C --> (SubGrp ` G ) -> dom ( S |` C ) = C )
36	34, 35	syl 17	. . . . . . . . 9 |- ( ph -> dom ( S |` C ) = C )
37	36	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. C ) -> dom ( S |` C ) = C )
38		simpr 477	. . . . . . . 8 |- ( ( ph /\ x e. C ) -> x e. C )
39	33, 37, 38	dprdub 18424	. . . . . . 7 |- ( ( ph /\ x e. C ) -> ( ( S |` C ) ` x ) C_ ( G DProd ( S |` C ) ) )
40	32, 39	eqsstr3d 3640	. . . . . 6 |- ( ( ph /\ x e. C ) -> ( S ` x ) C_ ( G DProd ( S |` C ) ) )
41	24	lsmub1 18071	. . . . . . . 8 |- ( ( ( G DProd ( S |` C ) ) e. (SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. (SubGrp ` G ) ) -> ( G DProd ( S |` C ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
42	11, 17, 41	syl2anc 693	. . . . . . 7 |- ( ph -> ( G DProd ( S |` C ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
43	42	adantr 481	. . . . . 6 |- ( ( ph /\ x e. C ) -> ( G DProd ( S |` C ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
44	40, 43	sstrd 3613	. . . . 5 |- ( ( ph /\ x e. C ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
45		fvres 6207	. . . . . . . 8 |- ( x e. D -> ( ( S |` D ) ` x ) = ( S ` x ) )
46	45	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. D ) -> ( ( S |` D ) ` x ) = ( S ` x ) )
47	15	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> G dom DProd ( S |` D ) )
48	2, 13	fssresd 6071	. . . . . . . . . 10 |- ( ph -> ( S |` D ) : D --> (SubGrp ` G ) )
49		fdm 6051	. . . . . . . . . 10 |- ( ( S |` D ) : D --> (SubGrp ` G ) -> dom ( S |` D ) = D )
50	48, 49	syl 17	. . . . . . . . 9 |- ( ph -> dom ( S |` D ) = D )
51	50	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> dom ( S |` D ) = D )
52		simpr 477	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> x e. D )
53	47, 51, 52	dprdub 18424	. . . . . . 7 |- ( ( ph /\ x e. D ) -> ( ( S |` D ) ` x ) C_ ( G DProd ( S |` D ) ) )
54	46, 53	eqsstr3d 3640	. . . . . 6 |- ( ( ph /\ x e. D ) -> ( S ` x ) C_ ( G DProd ( S |` D ) ) )
55	24	lsmub2 18072	. . . . . . . 8 |- ( ( ( G DProd ( S |` C ) ) e. (SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. (SubGrp ` G ) ) -> ( G DProd ( S |` D ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
56	11, 17, 55	syl2anc 693	. . . . . . 7 |- ( ph -> ( G DProd ( S |` D ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
57	56	adantr 481	. . . . . 6 |- ( ( ph /\ x e. D ) -> ( G DProd ( S |` D ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
58	54, 57	sstrd 3613	. . . . 5 |- ( ( ph /\ x e. D ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
59	44, 58	jaodan 826	. . . 4 |- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
60	30, 59	syldan 487	. . 3 |- ( ( ph /\ x e. I ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
61	1, 4, 26, 60	dprdlub 18425	. 2 |- ( ph -> ( G DProd S ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )
62	8	simprd 479	. . 3 |- ( ph -> ( G DProd ( S |` C ) ) C_ ( G DProd S ) )
63	14	simprd 479	. . 3 |- ( ph -> ( G DProd ( S |` D ) ) C_ ( G DProd S ) )
64		dprdsubg 18423	. . . . 5 |- ( G dom DProd S -> ( G DProd S ) e. (SubGrp ` G ) )
65	1, 64	syl 17	. . . 4 |- ( ph -> ( G DProd S ) e. (SubGrp ` G ) )
66	24	lsmlub 18078	. . . 4 |- ( ( ( G DProd ( S |` C ) ) e. (SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. (SubGrp ` G ) /\ ( G DProd S ) e. (SubGrp ` G ) ) -> ( ( ( G DProd ( S |` C ) ) C_ ( G DProd S ) /\ ( G DProd ( S |` D ) ) C_ ( G DProd S ) ) <-> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) C_ ( G DProd S ) ) )
67	11, 17, 65, 66	syl3anc 1326	. . 3 |- ( ph -> ( ( ( G DProd ( S |` C ) ) C_ ( G DProd S ) /\ ( G DProd ( S |` D ) ) C_ ( G DProd S ) ) <-> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) C_ ( G DProd S ) ) )
68	62, 63, 67	mpbi2and 956	. 2 |- ( ph -> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) C_ ( G DProd S ) )
69	61, 68	eqssd 3620	1 |- ( ph -> ( G DProd S ) = ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   0gc0g 16100  SubGrpcsubg 17588  Cntzccntz 17748   LSSumclsm 18049   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:  dprdpr  18449  dpjlsm  18453  ablfac1eulem  18471  ablfac1eu  18472  pgpfaclem1  18480