Theorem subgdprd 18434

Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)

Hypotheses

Ref	Expression
subgdprd.1	|- H = ( G ↾s A )
subgdprd.2	|- ( ph -> A e. (SubGrp ` G ) )
subgdprd.3	|- ( ph -> G dom DProd S )
subgdprd.4	|- ( ph -> ran S C_ ~P A )

Assertion

Ref	Expression
subgdprd	|- ( ph -> ( H DProd S ) = ( G DProd S ) )

Proof of Theorem subgdprd

Step	Hyp	Ref	Expression
1		subgdprd.2	. . . . . 6 |- ( ph -> A e. (SubGrp ` G ) )
2		subgdprd.1	. . . . . . 7 |- H = ( G ↾s A )
3	2	subggrp 17597	. . . . . 6 |- ( A e. (SubGrp ` G ) -> H e. Grp )
4	1, 3	syl 17	. . . . 5 |- ( ph -> H e. Grp )
5		eqid 2622	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
6	5	subgacs 17629	. . . . 5 |- ( H e. Grp -> (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) )
7		acsmre 16313	. . . . 5 |- ( (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
8	4, 6, 7	3syl 18	. . . 4 |- ( ph -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
9		subgrcl 17599	. . . . . . 7 |- ( A e. (SubGrp ` G ) -> G e. Grp )
10	1, 9	syl 17	. . . . . 6 |- ( ph -> G e. Grp )
11		eqid 2622	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
12	11	subgacs 17629	. . . . . 6 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
13		acsmre 16313	. . . . . 6 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
14	10, 12, 13	3syl 18	. . . . 5 |- ( ph -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
15		eqid 2622	. . . . 5 |- (mrCls ` (SubGrp ` G ) ) = (mrCls ` (SubGrp ` G ) )
16		subgdprd.3	. . . . . . . 8 |- ( ph -> G dom DProd S )
17		dprdf 18405	. . . . . . . 8 |- ( G dom DProd S -> S : dom S --> (SubGrp ` G ) )
18		frn 6053	. . . . . . . 8 |- ( S : dom S --> (SubGrp ` G ) -> ran S C_ (SubGrp ` G ) )
19	16, 17, 18	3syl 18	. . . . . . 7 |- ( ph -> ran S C_ (SubGrp ` G ) )
20		mresspw 16252	. . . . . . . 8 |- ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
21	14, 20	syl 17	. . . . . . 7 |- ( ph -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
22	19, 21	sstrd 3613	. . . . . 6 |- ( ph -> ran S C_ ~P ( Base ` G ) )
23		sspwuni 4611	. . . . . 6 |- ( ran S C_ ~P ( Base ` G ) <-> U. ran S C_ ( Base ` G ) )
24	22, 23	sylib 208	. . . . 5 |- ( ph -> U. ran S C_ ( Base ` G ) )
25	14, 15, 24	mrcssidd 16285	. . . 4 |- ( ph -> U. ran S C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
26	15	mrccl 16271	. . . . . 6 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ran S C_ ( Base ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` G ) )
27	14, 24, 26	syl2anc 693	. . . . 5 |- ( ph -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` G ) )
28		subgdprd.4	. . . . . . 7 |- ( ph -> ran S C_ ~P A )
29		sspwuni 4611	. . . . . . 7 |- ( ran S C_ ~P A <-> U. ran S C_ A )
30	28, 29	sylib 208	. . . . . 6 |- ( ph -> U. ran S C_ A )
31	15	mrcsscl 16280	. . . . . 6 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ran S C_ A /\ A e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) C_ A )
32	14, 30, 1, 31	syl3anc 1326	. . . . 5 |- ( ph -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) C_ A )
33	2	subsubg 17617	. . . . . 6 |- ( A e. (SubGrp ` G ) -> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) C_ A ) ) )
34	1, 33	syl 17	. . . . 5 |- ( ph -> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) C_ A ) ) )
35	27, 32, 34	mpbir2and 957	. . . 4 |- ( ph -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` H ) )
36		eqid 2622	. . . . 5 |- (mrCls ` (SubGrp ` H ) ) = (mrCls ` (SubGrp ` H ) )
37	36	mrcsscl 16280	. . . 4 |- ( ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) /\ U. ran S C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) /\ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) e. (SubGrp ` H ) ) -> ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
38	8, 25, 35, 37	syl3anc 1326	. . 3 |- ( ph -> ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
39	2	subgdmdprd 18433	. . . . . . . . . . 11 |- ( A e. (SubGrp ` G ) -> ( H dom DProd S <-> ( G dom DProd S /\ ran S C_ ~P A ) ) )
40	1, 39	syl 17	. . . . . . . . . 10 |- ( ph -> ( H dom DProd S <-> ( G dom DProd S /\ ran S C_ ~P A ) ) )
41	16, 28, 40	mpbir2and 957	. . . . . . . . 9 |- ( ph -> H dom DProd S )
42		eqidd 2623	. . . . . . . . 9 |- ( ph -> dom S = dom S )
43	41, 42	dprdf2 18406	. . . . . . . 8 |- ( ph -> S : dom S --> (SubGrp ` H ) )
44		frn 6053	. . . . . . . 8 |- ( S : dom S --> (SubGrp ` H ) -> ran S C_ (SubGrp ` H ) )
45	43, 44	syl 17	. . . . . . 7 |- ( ph -> ran S C_ (SubGrp ` H ) )
46		mresspw 16252	. . . . . . . 8 |- ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) -> (SubGrp ` H ) C_ ~P ( Base ` H ) )
47	8, 46	syl 17	. . . . . . 7 |- ( ph -> (SubGrp ` H ) C_ ~P ( Base ` H ) )
48	45, 47	sstrd 3613	. . . . . 6 |- ( ph -> ran S C_ ~P ( Base ` H ) )
49		sspwuni 4611	. . . . . 6 |- ( ran S C_ ~P ( Base ` H ) <-> U. ran S C_ ( Base ` H ) )
50	48, 49	sylib 208	. . . . 5 |- ( ph -> U. ran S C_ ( Base ` H ) )
51	8, 36, 50	mrcssidd 16285	. . . 4 |- ( ph -> U. ran S C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) )
52	36	mrccl 16271	. . . . . . 7 |- ( ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) /\ U. ran S C_ ( Base ` H ) ) -> ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` H ) )
53	8, 50, 52	syl2anc 693	. . . . . 6 |- ( ph -> ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` H ) )
54	2	subsubg 17617	. . . . . . 7 |- ( A e. (SubGrp ` G ) -> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) C_ A ) ) )
55	1, 54	syl 17	. . . . . 6 |- ( ph -> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) C_ A ) ) )
56	53, 55	mpbid 222	. . . . 5 |- ( ph -> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) C_ A ) )
57	56	simpld 475	. . . 4 |- ( ph -> ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` G ) )
58	15	mrcsscl 16280	. . . 4 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ran S C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) )
59	14, 51, 57, 58	syl3anc 1326	. . 3 |- ( ph -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) )
60	38, 59	eqssd 3620	. 2 |- ( ph -> ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
61	36	dprdspan 18426	. . 3 |- ( H dom DProd S -> ( H DProd S ) = ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) )
62	41, 61	syl 17	. 2 |- ( ph -> ( H DProd S ) = ( (mrCls ` (SubGrp ` H ) ) ` U. ran S ) )
63	15	dprdspan 18426	. . 3 |- ( G dom DProd S -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
64	16, 63	syl 17	. 2 |- ( ph -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
65	60, 62, 64	3eqtr4d 2666	1 |- ( ph -> ( H DProd S ) = ( G DProd S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Grpcgrp 17422  SubGrpcsubg 17588   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:  ablfaclem3  18486