Theorem dprdss 18428

Description: Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdss.1	|- ( ph -> G dom DProd T )
dprdss.2	|- ( ph -> dom T = I )
dprdss.3	|- ( ph -> S : I --> (SubGrp ` G ) )
dprdss.4	|- ( ( ph /\ k e. I ) -> ( S ` k ) C_ ( T ` k ) )

Assertion

Ref	Expression
dprdss	|- ( ph -> ( G dom DProd S /\ ( G DProd S ) C_ ( G DProd T ) ) )

Distinct variable groups:   k, G   ph, k   S, k   T, k   k, I

Proof of Theorem dprdss

Dummy variables f a h 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		dprdss.1	. . . 4 |- ( ph -> G dom DProd T )
5		dprdgrp 18404	. . . 4 |- ( G dom DProd T -> G e. Grp )
6	4, 5	syl 17	. . 3 |- ( ph -> G e. Grp )
7		dprdss.2	. . . 4 |- ( ph -> dom T = I )
8	4, 7	dprddomcld 18400	. . 3 |- ( ph -> I e. _V )
9		dprdss.3	. . 3 |- ( ph -> S : I --> (SubGrp ` G ) )
10		dprdss.4	. . . . . . 7 |- ( ( ph /\ k e. I ) -> ( S ` k ) C_ ( T ` k ) )
11	10	ralrimiva 2966	. . . . . 6 |- ( ph -> A. k e. I ( S ` k ) C_ ( T ` k ) )
12		fveq2 6191	. . . . . . . 8 |- ( k = x -> ( S ` k ) = ( S ` x ) )
13		fveq2 6191	. . . . . . . 8 |- ( k = x -> ( T ` k ) = ( T ` x ) )
14	12, 13	sseq12d 3634	. . . . . . 7 |- ( k = x -> ( ( S ` k ) C_ ( T ` k ) <-> ( S ` x ) C_ ( T ` x ) ) )
15	14	rspcv 3305	. . . . . 6 |- ( x e. I -> ( A. k e. I ( S ` k ) C_ ( T ` k ) -> ( S ` x ) C_ ( T ` x ) ) )
16	11, 15	mpan9 486	. . . . 5 |- ( ( ph /\ x e. I ) -> ( S ` x ) C_ ( T ` x ) )
17	16	3ad2antr1 1226	. . . 4 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( S ` x ) C_ ( T ` x ) )
18	4	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> G dom DProd T )
19	7	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> dom T = I )
20		simpr1 1067	. . . . . 6 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> x e. I )
21		simpr2 1068	. . . . . 6 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> y e. I )
22		simpr3 1069	. . . . . 6 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> x =/= y )
23	18, 19, 20, 21, 22, 1	dprdcntz 18407	. . . . 5 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( T ` x ) C_ ( (Cntz ` G ) ` ( T ` y ) ) )
24	4, 7	dprdf2 18406	. . . . . . . . 9 |- ( ph -> T : I --> (SubGrp ` G ) )
25	24	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> T : I --> (SubGrp ` G ) )
26	25, 21	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( T ` y ) e. (SubGrp ` G ) )
27		eqid 2622	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
28	27	subgss 17595	. . . . . . 7 |- ( ( T ` y ) e. (SubGrp ` G ) -> ( T ` y ) C_ ( Base ` G ) )
29	26, 28	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( T ` y ) C_ ( Base ` G ) )
30	11	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> A. k e. I ( S ` k ) C_ ( T ` k ) )
31		fveq2 6191	. . . . . . . . 9 |- ( k = y -> ( S ` k ) = ( S ` y ) )
32		fveq2 6191	. . . . . . . . 9 |- ( k = y -> ( T ` k ) = ( T ` y ) )
33	31, 32	sseq12d 3634	. . . . . . . 8 |- ( k = y -> ( ( S ` k ) C_ ( T ` k ) <-> ( S ` y ) C_ ( T ` y ) ) )
34	33	rspcv 3305	. . . . . . 7 |- ( y e. I -> ( A. k e. I ( S ` k ) C_ ( T ` k ) -> ( S ` y ) C_ ( T ` y ) ) )
35	21, 30, 34	sylc 65	. . . . . 6 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( S ` y ) C_ ( T ` y ) )
36	27, 1	cntz2ss 17765	. . . . . 6 |- ( ( ( T ` y ) C_ ( Base ` G ) /\ ( S ` y ) C_ ( T ` y ) ) -> ( (Cntz ` G ) ` ( T ` y ) ) C_ ( (Cntz ` G ) ` ( S ` y ) ) )
37	29, 35, 36	syl2anc 693	. . . . 5 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( (Cntz ` G ) ` ( T ` y ) ) C_ ( (Cntz ` G ) ` ( S ` y ) ) )
38	23, 37	sstrd 3613	. . . 4 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( T ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) )
39	17, 38	sstrd 3613	. . 3 |- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) )
40	6	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> G e. Grp )
41	27	subgacs 17629	. . . . . . 7 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
42		acsmre 16313	. . . . . . 7 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
43	40, 41, 42	3syl 18	. . . . . 6 |- ( ( ph /\ x e. I ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
44		difss 3737	. . . . . . . . 9 |- ( I \ { x } ) C_ I
45	11	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. I ) -> A. k e. I ( S ` k ) C_ ( T ` k ) )
46		ssralv 3666	. . . . . . . . 9 |- ( ( I \ { x } ) C_ I -> ( A. k e. I ( S ` k ) C_ ( T ` k ) -> A. k e. ( I \ { x } ) ( S ` k ) C_ ( T ` k ) ) )
47	44, 45, 46	mpsyl 68	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> A. k e. ( I \ { x } ) ( S ` k ) C_ ( T ` k ) )
48		ss2iun 4536	. . . . . . . 8 |- ( A. k e. ( I \ { x } ) ( S ` k ) C_ ( T ` k ) -> U_ k e. ( I \ { x } ) ( S ` k ) C_ U_ k e. ( I \ { x } ) ( T ` k ) )
49	47, 48	syl 17	. . . . . . 7 |- ( ( ph /\ x e. I ) -> U_ k e. ( I \ { x } ) ( S ` k ) C_ U_ k e. ( I \ { x } ) ( T ` k ) )
50	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> S : I --> (SubGrp ` G ) )
51		ffun 6048	. . . . . . . 8 |- ( S : I --> (SubGrp ` G ) -> Fun S )
52		funiunfv 6506	. . . . . . . 8 |- ( Fun S -> U_ k e. ( I \ { x } ) ( S ` k ) = U. ( S " ( I \ { x } ) ) )
53	50, 51, 52	3syl 18	. . . . . . 7 |- ( ( ph /\ x e. I ) -> U_ k e. ( I \ { x } ) ( S ` k ) = U. ( S " ( I \ { x } ) ) )
54	24	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> T : I --> (SubGrp ` G ) )
55		ffun 6048	. . . . . . . 8 |- ( T : I --> (SubGrp ` G ) -> Fun T )
56		funiunfv 6506	. . . . . . . 8 |- ( Fun T -> U_ k e. ( I \ { x } ) ( T ` k ) = U. ( T " ( I \ { x } ) ) )
57	54, 55, 56	3syl 18	. . . . . . 7 |- ( ( ph /\ x e. I ) -> U_ k e. ( I \ { x } ) ( T ` k ) = U. ( T " ( I \ { x } ) ) )
58	49, 53, 57	3sstr3d 3647	. . . . . 6 |- ( ( ph /\ x e. I ) -> U. ( S " ( I \ { x } ) ) C_ U. ( T " ( I \ { x } ) ) )
59		imassrn 5477	. . . . . . . 8 |- ( T " ( I \ { x } ) ) C_ ran T
60		frn 6053	. . . . . . . . . 10 |- ( T : I --> (SubGrp ` G ) -> ran T C_ (SubGrp ` G ) )
61	54, 60	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. I ) -> ran T C_ (SubGrp ` G ) )
62		mresspw 16252	. . . . . . . . . 10 |- ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
63	43, 62	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. I ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
64	61, 63	sstrd 3613	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> ran T C_ ~P ( Base ` G ) )
65	59, 64	syl5ss 3614	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( T " ( I \ { x } ) ) C_ ~P ( Base ` G ) )
66		sspwuni 4611	. . . . . . 7 |- ( ( T " ( I \ { x } ) ) C_ ~P ( Base ` G ) <-> U. ( T " ( I \ { x } ) ) C_ ( Base ` G ) )
67	65, 66	sylib 208	. . . . . 6 |- ( ( ph /\ x e. I ) -> U. ( T " ( I \ { x } ) ) C_ ( Base ` G ) )
68	43, 3, 58, 67	mrcssd 16284	. . . . 5 |- ( ( ph /\ x e. I ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( T " ( I \ { x } ) ) ) )
69		ss2in 3840	. . . . 5 |- ( ( ( S ` x ) C_ ( T ` x ) /\ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( T " ( I \ { x } ) ) ) ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ ( ( T ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( T " ( I \ { x } ) ) ) ) )
70	16, 68, 69	syl2anc 693	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ ( ( T ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( T " ( I \ { x } ) ) ) ) )
71	4	adantr 481	. . . . 5 |- ( ( ph /\ x e. I ) -> G dom DProd T )
72	7	adantr 481	. . . . 5 |- ( ( ph /\ x e. I ) -> dom T = I )
73		simpr 477	. . . . 5 |- ( ( ph /\ x e. I ) -> x e. I )
74	71, 72, 73, 2, 3	dprddisj 18408	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( T ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( T " ( I \ { x } ) ) ) ) = { ( 0g ` G ) } )
75	70, 74	sseqtrd 3641	. . 3 |- ( ( ph /\ x e. I ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { ( 0g ` G ) } )
76	1, 2, 3, 6, 8, 9, 39, 75	dmdprdd 18398	. 2 |- ( ph -> G dom DProd S )
77	4	a1d 25	. . . . 5 |- ( ph -> ( G dom DProd S -> G dom DProd T ) )
78		ss2ixp 7921	. . . . . . 7 |- ( A. k e. I ( S ` k ) C_ ( T ` k ) -> X_ k e. I ( S ` k ) C_ X_ k e. I ( T ` k ) )
79	11, 78	syl 17	. . . . . 6 |- ( ph -> X_ k e. I ( S ` k ) C_ X_ k e. I ( T ` k ) )
80		rabss2 3685	. . . . . 6 |- ( X_ k e. I ( S ` k ) C_ X_ k e. I ( T ` k ) -> { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } C_ { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } )
81		ssrexv 3667	. . . . . 6 |- ( { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } C_ { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } -> ( E. f e. { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) -> E. f e. { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) )
82	79, 80, 81	3syl 18	. . . . 5 |- ( ph -> ( E. f e. { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) -> E. f e. { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) )
83	77, 82	anim12d 586	. . . 4 |- ( ph -> ( ( G dom DProd S /\ E. f e. { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) -> ( G dom DProd T /\ E. f e. { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) ) )
84		fdm 6051	. . . . 5 |- ( S : I --> (SubGrp ` G ) -> dom S = I )
85		eqid 2622	. . . . . 6 |- { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } = { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) }
86	2, 85	eldprd 18403	. . . . 5 |- ( dom S = I -> ( a e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) ) )
87	9, 84, 86	3syl 18	. . . 4 |- ( ph -> ( a e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. { h e. X_ k e. I ( S ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) ) )
88		eqid 2622	. . . . . 6 |- { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } = { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) }
89	2, 88	eldprd 18403	. . . . 5 |- ( dom T = I -> ( a e. ( G DProd T ) <-> ( G dom DProd T /\ E. f e. { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) ) )
90	7, 89	syl 17	. . . 4 |- ( ph -> ( a e. ( G DProd T ) <-> ( G dom DProd T /\ E. f e. { h e. X_ k e. I ( T ` k ) | h finSupp ( 0g ` G ) } a = ( G gsumg f ) ) ) )
91	83, 87, 90	3imtr4d 283	. . 3 |- ( ph -> ( a e. ( G DProd S ) -> a e. ( G DProd T ) ) )
92	91	ssrdv 3609	. 2 |- ( ph -> ( G DProd S ) C_ ( G DProd T ) )
93	76, 92	jca 554	1 |- ( ph -> ( G dom DProd S /\ ( G DProd S ) C_ ( G DProd T ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   U_ciun 4520   class class class wbr 4653   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   X_cixp 7908   finSupp cfsupp 8275   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101  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-cntz 17750  df-dprd 18394

This theorem is referenced by: (None)