Theorem subgdprd 20131

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 19219	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	syl 18	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ Grp)
5		eqid 2765	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
6	5	subgacs 19251	. . . . 5 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7		acsmre 17726	. . . . 5 ⊢ ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
8	4, 6, 7	3syl 19	. . . 4 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
9		subgrcl 19221	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
10	1, 9	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
12	11	subgacs 19251	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13		acsmre 17726	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
14	10, 12, 13	3syl 19	. . . . 5 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
15		eqid 2765	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16		subgdprd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
17		dprdf 20102	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18		frn 6717	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
19	16, 17, 18	3syl 19	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20		mresspw 17662	. . . . . . . 8 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
21	14, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
22	19, 21	sstrd 3948	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23		sspwuni 5068	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺))
24	22, 23	sylib 221	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐺))
25	14, 15, 24	mrcssidd 17699	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
26	15	mrccl 17685	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
27	14, 24, 26	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
28		subgdprd.4	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29		sspwuni 5068	. . . . . . 7 ⊢ (ran 𝑆 ⊆ 𝒫 𝐴 ↔ ∪ ran 𝑆 ⊆ 𝐴)
30	28, 29	sylib 221	. . . . . 6 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ 𝐴)
31	15	mrcsscl 17694	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ 𝐴 ∧ 𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
32	14, 30, 1, 31	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
33	2	subsubg 19240	. . . . . 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 18	. . . . 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 726	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
36		eqid 2765	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
37	36	mrcsscl 17694	. . . 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 1398	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
39	2	subgdmdprd 20130	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
40	1, 39	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
41	16, 28, 40	mpbir2and 726	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
42		eqidd 2766	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑆 = dom 𝑆)
43	41, 42	dprdf2 20103	. . . . . . . 8 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
44	43	frnd 6718	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45		mresspw 17662	. . . . . . . 8 ⊢ ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
46	8, 45	syl 18	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
47	44, 46	sstrd 3948	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48		sspwuni 5068	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐻))
49	47, 48	sylib 221	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐻))
50	8, 36, 49	mrcssidd 17699	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
51	36	mrccl 17685	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
52	8, 49, 51	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
53	2	subsubg 19240	. . . . . . 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 18	. . . . . 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 235	. . . . 5 ⊢ (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ 𝐴))
56	55	simpld 500	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
57	15	mrcsscl 17694	. . . 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 1398	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
59	38, 58	eqssd 3955	. 2 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
60	36	dprdspan 20123	. . 3 ⊢ (𝐻dom DProd 𝑆 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
61	41, 60	syl 18	. 2 ⊢ (𝜑 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
62	15	dprdspan 20123	. . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
63	16, 62	syl 18	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
64	59, 61, 63	3eqtr4d 2810	1 ⊢ (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  𝒫 cpw 4564  ∪ cuni 4874   class class class wbr 5111  dom cdm 5663  ran crn 5664  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  Basecbs 17287   ↾_s cress 17308  Moorecmre 17652  mrClscmrc 17653  ACScacs 17655  Grpcgrp 19024  SubGrpcsubg 19210   DProd cdprd 20089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7680  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9325  df-oi 9475  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-n0 12516  df-z 12603  df-uz 12875  df-fz 13548  df-fzo 13696  df-seq 14052  df-hash 14381  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-ress 17309  df-plusg 17341  df-0g 17512  df-gsum 17513  df-mre 17656  df-mrc 17657  df-acs 17659  df-mgm 18716  df-sgrp 18799  df-mnd 18815  df-mhm 18865  df-submnd 18866  df-grp 19027  df-minusg 19028  df-sbg 19029  df-mulg 19158  df-subg 19213  df-ghm 19308  df-gim 19353  df-cntz 19411  df-oppg 19440  df-cmn 19876  df-dprd 20091

This theorem is used by:  ablfaclem3  20183