Theorem subgdprd 20003

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 19096	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ Grp)
5		eqid 2739	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
6	5	subgacs 19127	. . . . 5 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7		acsmre 17609	. . . . 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 19098	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		eqid 2739	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
12	11	subgacs 19127	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13		acsmre 17609	. . . . . 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 2739	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16		subgdprd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
17		dprdf 19974	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18		frn 6662	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20		mresspw 17545	. . . . . . . 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 3925	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23		sspwuni 5029	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺))
24	22, 23	sylib 219	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐺))
25	14, 15, 24	mrcssidd 17582	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
26	15	mrccl 17568	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
27	14, 24, 26	syl2anc 590	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
28		subgdprd.4	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29		sspwuni 5029	. . . . . . 7 ⊢ (ran 𝑆 ⊆ 𝒫 𝐴 ↔ ∪ ran 𝑆 ⊆ 𝐴)
30	28, 29	sylib 219	. . . . . 6 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ 𝐴)
31	15	mrcsscl 17577	. . . . . 6 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ 𝐴 ∧ 𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
32	14, 30, 1, 31	syl3anc 1379	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
33	2	subsubg 19116	. . . . . 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 719	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
36		eqid 2739	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
37	36	mrcsscl 17577	. . . 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 1379	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
39	2	subgdmdprd 20002	. . . . . . . . . . 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 719	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
42		eqidd 2740	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑆 = dom 𝑆)
43	41, 42	dprdf2 19975	. . . . . . . 8 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
44	43	frnd 6663	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45		mresspw 17545	. . . . . . . 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 3925	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48		sspwuni 5029	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐻))
49	47, 48	sylib 219	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐻))
50	8, 36, 49	mrcssidd 17582	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
51	36	mrccl 17568	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
52	8, 49, 51	syl2anc 590	. . . . . 6 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
53	2	subsubg 19116	. . . . . . 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 233	. . . . 5 ⊢ (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ 𝐴))
56	55	simpld 495	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
57	15	mrcsscl 17577	. . . 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 1379	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
59	38, 58	eqssd 3932	. 2 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
60	36	dprdspan 19995	. . 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 19995	. . 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 2784	1 ⊢ (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  𝒫 cpw 4529  ∪ cuni 4838   class class class wbr 5072  dom cdm 5618  ran crn 5619  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Basecbs 17170   ↾_s cress 17191  Moorecmre 17535  mrClscmrc 17536  ACScacs 17538  Grpcgrp 18900  SubGrpcsubg 19087   DProd cdprd 19961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-0g 17395  df-gsum 17396  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-ghm 19179  df-gim 19225  df-cntz 19283  df-oppg 19312  df-cmn 19748  df-dprd 19963

This theorem is referenced by:  ablfaclem3  20055