Theorem subgdprd 18897

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	⊢ (𝜑 → 𝐴 ∈ (SubGrp‘𝐺))
subgdprd.3	⊢ (𝜑 → 𝐺dom DProd 𝑆)
subgdprd.4	⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)

Assertion

Ref	Expression
subgdprd	⊢ (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

Proof of Theorem subgdprd

Step	Hyp	Ref	Expression
1		subgdprd.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (SubGrp‘𝐺))
2		subgdprd.1	. . . . . . 7 ⊢ 𝐻 = (𝐺 ↾s 𝐴)
3	2	subggrp 18056	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ Grp)
5		eqid 2772	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
6	5	subgacs 18088	. . . . 5 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7		acsmre 16771	. . . . 5 ⊢ ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
8	4, 6, 7	3syl 18	. . . 4 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
9		subgrcl 18058	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		eqid 2772	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
12	11	subgacs 18088	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13		acsmre 16771	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
14	10, 12, 13	3syl 18	. . . . 5 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
15		eqid 2772	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16		subgdprd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
17		dprdf 18868	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18		frn 6344	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20		mresspw 16711	. . . . . . . 8 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
21	14, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
22	19, 21	sstrd 3864	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23		sspwuni 4882	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺))
24	22, 23	sylib 210	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐺))
25	14, 15, 24	mrcssidd 16744	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
26	15	mrccl 16730	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
27	14, 24, 26	syl2anc 576	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
28		subgdprd.4	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29		sspwuni 4882	. . . . . . 7 ⊢ (ran 𝑆 ⊆ 𝒫 𝐴 ↔ ∪ ran 𝑆 ⊆ 𝐴)
30	28, 29	sylib 210	. . . . . 6 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ 𝐴)
31	15	mrcsscl 16739	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ 𝐴 ∧ 𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
32	14, 30, 1, 31	syl3anc 1351	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
33	2	subsubg 18076	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)))
34	1, 33	syl 17	. . . . 5 ⊢ (𝜑 → (((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)))
35	27, 32, 34	mpbir2and 700	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
36		eqid 2772	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
37	36	mrcsscl 16739	. . . 4 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
38	8, 25, 35, 37	syl3anc 1351	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
39	2	subgdmdprd 18896	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
40	1, 39	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
41	16, 28, 40	mpbir2and 700	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
42		eqidd 2773	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑆 = dom 𝑆)
43	41, 42	dprdf2 18869	. . . . . . . 8 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
44	43	frnd 6345	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45		mresspw 16711	. . . . . . . 8 ⊢ ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
46	8, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
47	44, 46	sstrd 3864	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48		sspwuni 4882	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐻))
49	47, 48	sylib 210	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐻))
50	8, 36, 49	mrcssidd 16744	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
51	36	mrccl 16730	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
52	8, 49, 51	syl2anc 576	. . . . . 6 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
53	2	subsubg 18076	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ 𝐴)))
54	1, 53	syl 17	. . . . . 6 ⊢ (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ 𝐴)))
55	52, 54	mpbid 224	. . . . 5 ⊢ (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ 𝐴))
56	55	simpld 487	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
57	15	mrcsscl 16739	. . . 4 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
58	14, 50, 56, 57	syl3anc 1351	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
59	38, 58	eqssd 3871	. 2 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
60	36	dprdspan 18889	. . 3 ⊢ (𝐻dom DProd 𝑆 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
61	41, 60	syl 17	. 2 ⊢ (𝜑 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
62	15	dprdspan 18889	. . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
63	16, 62	syl 17	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
64	59, 61, 63	3eqtr4d 2818	1 ⊢ (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048   ⊆ wss 3825  𝒫 cpw 4416  ∪ cuni 4706   class class class wbr 4923  dom cdm 5400  ran crn 5401  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970  Basecbs 16329   ↾_s cress 16330  Moorecmre 16701  mrClscmrc 16702  ACScacs 16704  Grpcgrp 17881  SubGrpcsubg 18047   DProd cdprd 18855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7494  df-2nd 7495  df-supp 7627  df-tpos 7688  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-map 8200  df-ixp 8252  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fsupp 8621  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-n0 11701  df-z 11787  df-uz 12052  df-fz 12702  df-fzo 12843  df-seq 13178  df-hash 13499  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-0g 16561  df-gsum 16562  df-mre 16705  df-mrc 16706  df-acs 16708  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-mhm 17793  df-submnd 17794  df-grp 17884  df-minusg 17885  df-sbg 17886  df-mulg 18002  df-subg 18050  df-ghm 18117  df-gim 18160  df-cntz 18208  df-oppg 18235  df-cmn 18658  df-dprd 18857

This theorem is referenced by:  ablfaclem3  18949