Theorem dprdsn 18435

Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)

Assertion

Ref	Expression
dprdsn	|- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( G dom DProd { <. A , S >. } /\ ( G DProd { <. A , S >. } ) = S ) )

Proof of Theorem dprdsn

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		subgrcl 17599	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
5	4	adantl 482	. . 3 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> G e. Grp )
6		snex 4908	. . . 4 |- { A } e. _V
7	6	a1i 11	. . 3 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> { A } e. _V )
8		f1osng 6177	. . . . 5 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> { <. A , S >. } : { A } -1-1-onto-> { S } )
9		f1of 6137	. . . . 5 |- ( { <. A , S >. } : { A } -1-1-onto-> { S } -> { <. A , S >. } : { A } --> { S } )
10	8, 9	syl 17	. . . 4 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> { <. A , S >. } : { A } --> { S } )
11		simpr 477	. . . . 5 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> S e. (SubGrp ` G ) )
12	11	snssd 4340	. . . 4 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> { S } C_ (SubGrp ` G ) )
13	10, 12	fssd 6057	. . 3 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> { <. A , S >. } : { A } --> (SubGrp ` G ) )
14		simpr1 1067	. . . . . 6 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ ( x e. { A } /\ y e. { A } /\ x =/= y ) ) -> x e. { A } )
15		elsni 4194	. . . . . 6 |- ( x e. { A } -> x = A )
16	14, 15	syl 17	. . . . 5 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ ( x e. { A } /\ y e. { A } /\ x =/= y ) ) -> x = A )
17		simpr2 1068	. . . . . 6 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ ( x e. { A } /\ y e. { A } /\ x =/= y ) ) -> y e. { A } )
18		elsni 4194	. . . . . 6 |- ( y e. { A } -> y = A )
19	17, 18	syl 17	. . . . 5 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ ( x e. { A } /\ y e. { A } /\ x =/= y ) ) -> y = A )
20	16, 19	eqtr4d 2659	. . . 4 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ ( x e. { A } /\ y e. { A } /\ x =/= y ) ) -> x = y )
21		simpr3 1069	. . . 4 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ ( x e. { A } /\ y e. { A } /\ x =/= y ) ) -> x =/= y )
22	20, 21	pm2.21ddne 2878	. . 3 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ ( x e. { A } /\ y e. { A } /\ x =/= y ) ) -> ( { <. A , S >. } ` x ) C_ ( (Cntz ` G ) ` ( { <. A , S >. } ` y ) ) )
23	5	adantr 481	. . . . . . . 8 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> G e. Grp )
24		eqid 2622	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
25	24	subgacs 17629	. . . . . . . 8 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
26		acsmre 16313	. . . . . . . 8 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
27	23, 25, 26	3syl 18	. . . . . . 7 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
28	15	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> x = A )
29	28	sneqd 4189	. . . . . . . . . . . . . 14 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> { x } = { A } )
30	29	difeq2d 3728	. . . . . . . . . . . . 13 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( { A } \ { x } ) = ( { A } \ { A } ) )
31		difid 3948	. . . . . . . . . . . . 13 |- ( { A } \ { A } ) = (/)
32	30, 31	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( { A } \ { x } ) = (/) )
33	32	imaeq2d 5466	. . . . . . . . . . 11 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( { <. A , S >. } " ( { A } \ { x } ) ) = ( { <. A , S >. } " (/) ) )
34		ima0 5481	. . . . . . . . . . 11 |- ( { <. A , S >. } " (/) ) = (/)
35	33, 34	syl6eq 2672	. . . . . . . . . 10 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( { <. A , S >. } " ( { A } \ { x } ) ) = (/) )
36	35	unieqd 4446	. . . . . . . . 9 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> U. ( { <. A , S >. } " ( { A } \ { x } ) ) = U. (/) )
37		uni0 4465	. . . . . . . . 9 |- U. (/) = (/)
38	36, 37	syl6eq 2672	. . . . . . . 8 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> U. ( { <. A , S >. } " ( { A } \ { x } ) ) = (/) )
39		0ss 3972	. . . . . . . . 9 |- (/) C_ { ( 0g ` G ) }
40	39	a1i 11	. . . . . . . 8 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> (/) C_ { ( 0g ` G ) } )
41	38, 40	eqsstrd 3639	. . . . . . 7 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> U. ( { <. A , S >. } " ( { A } \ { x } ) ) C_ { ( 0g ` G ) } )
42	2	0subg 17619	. . . . . . . 8 |- ( G e. Grp -> { ( 0g ` G ) } e. (SubGrp ` G ) )
43	23, 42	syl 17	. . . . . . 7 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> { ( 0g ` G ) } e. (SubGrp ` G ) )
44	3	mrcsscl 16280	. . . . . . 7 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( { <. A , S >. } " ( { A } \ { x } ) ) C_ { ( 0g ` G ) } /\ { ( 0g ` G ) } e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) C_ { ( 0g ` G ) } )
45	27, 41, 43, 44	syl3anc 1326	. . . . . 6 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) C_ { ( 0g ` G ) } )
46	2	subg0cl 17602	. . . . . . . . 9 |- ( S e. (SubGrp ` G ) -> ( 0g ` G ) e. S )
47	46	ad2antlr 763	. . . . . . . 8 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( 0g ` G ) e. S )
48	15	fveq2d 6195	. . . . . . . . 9 |- ( x e. { A } -> ( { <. A , S >. } ` x ) = ( { <. A , S >. } ` A ) )
49		fvsng 6447	. . . . . . . . 9 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( { <. A , S >. } ` A ) = S )
50	48, 49	sylan9eqr 2678	. . . . . . . 8 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( { <. A , S >. } ` x ) = S )
51	47, 50	eleqtrrd 2704	. . . . . . 7 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( 0g ` G ) e. ( { <. A , S >. } ` x ) )
52	51	snssd 4340	. . . . . 6 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> { ( 0g ` G ) } C_ ( { <. A , S >. } ` x ) )
53	45, 52	sstrd 3613	. . . . 5 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) C_ ( { <. A , S >. } ` x ) )
54		sseqin2 3817	. . . . 5 |- ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) C_ ( { <. A , S >. } ` x ) <-> ( ( { <. A , S >. } ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) )
55	53, 54	sylib 208	. . . 4 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( ( { <. A , S >. } ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) )
56	55, 45	eqsstrd 3639	. . 3 |- ( ( ( A e. V /\ S e. (SubGrp ` G ) ) /\ x e. { A } ) -> ( ( { <. A , S >. } ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( { <. A , S >. } " ( { A } \ { x } ) ) ) ) C_ { ( 0g ` G ) } )
57	1, 2, 3, 5, 7, 13, 22, 56	dmdprdd 18398	. 2 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> G dom DProd { <. A , S >. } )
58	3	dprdspan 18426	. . . 4 |- ( G dom DProd { <. A , S >. } -> ( G DProd { <. A , S >. } ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran { <. A , S >. } ) )
59	57, 58	syl 17	. . 3 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( G DProd { <. A , S >. } ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran { <. A , S >. } ) )
60		rnsnopg 5614	. . . . . . . 8 |- ( A e. V -> ran { <. A , S >. } = { S } )
61	60	adantr 481	. . . . . . 7 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ran { <. A , S >. } = { S } )
62	61	unieqd 4446	. . . . . 6 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> U. ran { <. A , S >. } = U. { S } )
63		unisng 4452	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> U. { S } = S )
64	63	adantl 482	. . . . . 6 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> U. { S } = S )
65	62, 64	eqtrd 2656	. . . . 5 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> U. ran { <. A , S >. } = S )
66	65	fveq2d 6195	. . . 4 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran { <. A , S >. } ) = ( (mrCls ` (SubGrp ` G ) ) ` S ) )
67	5, 25, 26	3syl 18	. . . . 5 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
68	3	mrcid 16273	. . . . 5 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ S e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` S ) = S )
69	67, 68	sylancom 701	. . . 4 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` S ) = S )
70	66, 69	eqtrd 2656	. . 3 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran { <. A , S >. } ) = S )
71	59, 70	eqtrd 2656	. 2 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( G DProd { <. A , S >. } ) = S )
72	57, 71	jca 554	1 |- ( ( A e. V /\ S e. (SubGrp ` G ) ) -> ( G dom DProd { <. A , S >. } /\ ( G DProd { <. A , S >. } ) = S ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   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:  dprd2da  18441  dmdprdpr  18448  dprdpr  18449  dpjlem  18450  pgpfaclem1  18480