Theorem subgdprd 20007

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 19100	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ Grp)
5		eqid 2741	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
6	5	subgacs 19131	. . . . 5 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7		acsmre 17613	. . . . 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 19102	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		eqid 2741	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
12	11	subgacs 19131	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13		acsmre 17613	. . . . . 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 2741	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16		subgdprd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
17		dprdf 19978	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18		frn 6666	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20		mresspw 17549	. . . . . . . 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 3927	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23		sspwuni 5032	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺))
24	22, 23	sylib 220	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐺))
25	14, 15, 24	mrcssidd 17586	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
26	15	mrccl 17572	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
27	14, 24, 26	syl2anc 591	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
28		subgdprd.4	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29		sspwuni 5032	. . . . . . 7 ⊢ (ran 𝑆 ⊆ 𝒫 𝐴 ↔ ∪ ran 𝑆 ⊆ 𝐴)
30	28, 29	sylib 220	. . . . . 6 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ 𝐴)
31	15	mrcsscl 17581	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ 𝐴 ∧ 𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
32	14, 30, 1, 31	syl3anc 1380	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
33	2	subsubg 19120	. . . . . 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 720	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
36		eqid 2741	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
37	36	mrcsscl 17581	. . . 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 1380	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
39	2	subgdmdprd 20006	. . . . . . . . . . 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 720	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
42		eqidd 2742	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑆 = dom 𝑆)
43	41, 42	dprdf2 19979	. . . . . . . 8 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
44	43	frnd 6667	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45		mresspw 17549	. . . . . . . 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 3927	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48		sspwuni 5032	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐻))
49	47, 48	sylib 220	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐻))
50	8, 36, 49	mrcssidd 17586	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
51	36	mrccl 17572	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
52	8, 49, 51	syl2anc 591	. . . . . 6 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
53	2	subsubg 19120	. . . . . . 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 234	. . . . 5 ⊢ (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ 𝐴))
56	55	simpld 496	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
57	15	mrcsscl 17581	. . . 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 1380	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
59	38, 58	eqssd 3934	. 2 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
60	36	dprdspan 19999	. . 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 19999	. . 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 2786	1 ⊢ (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ⊆ wss 3885  𝒫 cpw 4532  ∪ cuni 4841   class class class wbr 5075  dom cdm 5621  ran crn 5622  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  Basecbs 17174   ↾_s cress 17195  Moorecmre 17539  mrClscmrc 17540  ACScacs 17542  Grpcgrp 18904  SubGrpcsubg 19091   DProd cdprd 19965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-oi 9419  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-hash 14288  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-0g 17399  df-gsum 17400  df-mre 17543  df-mrc 17544  df-acs 17546  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18746  df-submnd 18747  df-grp 18907  df-minusg 18908  df-sbg 18909  df-mulg 19039  df-subg 19094  df-ghm 19183  df-gim 19229  df-cntz 19287  df-oppg 19316  df-cmn 19752  df-dprd 19967

This theorem is referenced by:  ablfaclem3  20059