Theorem dmdprdd 18398

Description: Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dmdprd.z	|- Z = (Cntz ` G )
dmdprd.0	|- .0. = ( 0g ` G )
dmdprd.k	|- K = (mrCls ` (SubGrp ` G ) )
dmdprdd.1	|- ( ph -> G e. Grp )
dmdprdd.2	|- ( ph -> I e. V )
dmdprdd.3	|- ( ph -> S : I --> (SubGrp ` G ) )
dmdprdd.4	|- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( S ` x ) C_ ( Z ` ( S ` y ) ) )
dmdprdd.5	|- ( ( ph /\ x e. I ) -> ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } )

Assertion

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

Distinct variable groups:   x, y, G   x, I, y   ph, x, y   x, S, y   x, V, y

Allowed substitution hints:   K( x, y)   .0. ( x, y)   Z( x, y)

Proof of Theorem dmdprdd

Step	Hyp	Ref	Expression
1		dmdprdd.1	. 2 |- ( ph -> G e. Grp )
2		dmdprdd.3	. 2 |- ( ph -> S : I --> (SubGrp ` G ) )
3		eldifsn 4317	. . . . . . 7 |- ( y e. ( I \ { x } ) <-> ( y e. I /\ y =/= x ) )
4		necom 2847	. . . . . . . 8 |- ( y =/= x <-> x =/= y )
5	4	anbi2i 730	. . . . . . 7 |- ( ( y e. I /\ y =/= x ) <-> ( y e. I /\ x =/= y ) )
6	3, 5	bitri 264	. . . . . 6 |- ( y e. ( I \ { x } ) <-> ( y e. I /\ x =/= y ) )
7		dmdprdd.4	. . . . . . . 8 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( S ` x ) C_ ( Z ` ( S ` y ) ) )
8	7	3exp2 1285	. . . . . . 7 |- ( ph -> ( x e. I -> ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) ) )
9	8	imp4b 613	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( y e. I /\ x =/= y ) -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) )
10	6, 9	syl5bi 232	. . . . 5 |- ( ( ph /\ x e. I ) -> ( y e. ( I \ { x } ) -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) )
11	10	ralrimiv 2965	. . . 4 |- ( ( ph /\ x e. I ) -> A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) )
12		dmdprdd.5	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } )
13	2	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
14		dmdprd.0	. . . . . . . . 9 |- .0. = ( 0g ` G )
15	14	subg0cl 17602	. . . . . . . 8 |- ( ( S ` x ) e. (SubGrp ` G ) -> .0. e. ( S ` x ) )
16	13, 15	syl 17	. . . . . . 7 |- ( ( ph /\ x e. I ) -> .0. e. ( S ` x ) )
17	1	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. I ) -> G e. Grp )
18		eqid 2622	. . . . . . . . . . 11 |- ( Base ` G ) = ( Base ` G )
19	18	subgacs 17629	. . . . . . . . . 10 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
20		acsmre 16313	. . . . . . . . . 10 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
21	17, 19, 20	3syl 18	. . . . . . . . 9 |- ( ( ph /\ x e. I ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
22		imassrn 5477	. . . . . . . . . . . 12 |- ( S " ( I \ { x } ) ) C_ ran S
23		frn 6053	. . . . . . . . . . . . . 14 |- ( S : I --> (SubGrp ` G ) -> ran S C_ (SubGrp ` G ) )
24	2, 23	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ran S C_ (SubGrp ` G ) )
25	24	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. I ) -> ran S C_ (SubGrp ` G ) )
26	22, 25	syl5ss 3614	. . . . . . . . . . 11 |- ( ( ph /\ x e. I ) -> ( S " ( I \ { x } ) ) C_ (SubGrp ` G ) )
27		mresspw 16252	. . . . . . . . . . . 12 |- ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
28	21, 27	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ x e. I ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
29	26, 28	sstrd 3613	. . . . . . . . . 10 |- ( ( ph /\ x e. I ) -> ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) )
30		sspwuni 4611	. . . . . . . . . 10 |- ( ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) <-> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
31	29, 30	sylib 208	. . . . . . . . 9 |- ( ( ph /\ x e. I ) -> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
32		dmdprd.k	. . . . . . . . . 10 |- K = (mrCls ` (SubGrp ` G ) )
33	32	mrccl 16271	. . . . . . . . 9 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) ) -> ( K ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) )
34	21, 31, 33	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> ( K ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) )
35	14	subg0cl 17602	. . . . . . . 8 |- ( ( K ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) -> .0. e. ( K ` U. ( S " ( I \ { x } ) ) ) )
36	34, 35	syl 17	. . . . . . 7 |- ( ( ph /\ x e. I ) -> .0. e. ( K ` U. ( S " ( I \ { x } ) ) ) )
37	16, 36	elind 3798	. . . . . 6 |- ( ( ph /\ x e. I ) -> .0. e. ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) )
38	37	snssd 4340	. . . . 5 |- ( ( ph /\ x e. I ) -> { .0. } C_ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) )
39	12, 38	eqssd 3620	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } )
40	11, 39	jca 554	. . 3 |- ( ( ph /\ x e. I ) -> ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) )
41	40	ralrimiva 2966	. 2 |- ( ph -> A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) )
42		dmdprdd.2	. . 3 |- ( ph -> I e. V )
43		fdm 6051	. . . 4 |- ( S : I --> (SubGrp ` G ) -> dom S = I )
44	2, 43	syl 17	. . 3 |- ( ph -> dom S = I )
45		dmdprd.z	. . . 4 |- Z = (Cntz ` G )
46	45, 14, 32	dmdprd 18397	. . 3 |- ( ( I e. V /\ dom S = I ) -> ( G dom DProd S <-> ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
47	42, 44, 46	syl2anc 693	. 2 |- ( ph -> ( G dom DProd S <-> ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
48	1, 2, 41, 47	mpbir3and 1245	1 |- ( ph -> G dom DProd S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888   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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-dprd 18394

This theorem is referenced by:  dprdss  18428  dprdz  18429  dprdf1o  18431  dprdsn  18435  dprd2da  18441  dmdprdsplit2  18445  ablfac1b  18469