Theorem subgdprd 19967

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 19061	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ Grp)
5		eqid 2729	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
6	5	subgacs 19093	. . . . 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 19063	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
12	11	subgacs 19093	. . . . . 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 2729	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16		subgdprd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
17		dprdf 19938	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18		frn 6695	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20		mresspw 17553	. . . . . . . 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 3957	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23		sspwuni 5064	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺))
24	22, 23	sylib 218	. . . . 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 584	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺))
28		subgdprd.4	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29		sspwuni 5064	. . . . . . 7 ⊢ (ran 𝑆 ⊆ 𝒫 𝐴 ↔ ∪ ran 𝑆 ⊆ 𝐴)
30	28, 29	sylib 218	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ 𝐴)
33	2	subsubg 19081	. . . . . 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 713	. . . 4 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
36		eqid 2729	. . . . 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 1373	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
39	2	subgdmdprd 19966	. . . . . . . . . . 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 713	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
42		eqidd 2730	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑆 = dom 𝑆)
43	41, 42	dprdf2 19939	. . . . . . . 8 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
44	43	frnd 6696	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45		mresspw 17553	. . . . . . . 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 3957	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48		sspwuni 5064	. . . . . 6 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐻))
49	47, 48	sylib 218	. . . . 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 584	. . . . . 6 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐻))
53	2	subsubg 19081	. . . . . . 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 232	. . . . 5 ⊢ (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) ⊆ 𝐴))
56	55	simpld 494	. . . 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 1373	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆))
59	38, 58	eqssd 3964	. 2 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘∪ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
60	36	dprdspan 19959	. . 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 19959	. . 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 2774	1 ⊢ (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871   class class class wbr 5107  dom cdm 5638  ran crn 5639  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  Moorecmre 17543  mrClscmrc 17544  ACScacs 17546  Grpcgrp 18865  SubGrpcsubg 19052   DProd cdprd 19925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-gsum 17405  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-gim 19191  df-cntz 19249  df-oppg 19278  df-cmn 19712  df-dprd 19927

This theorem is referenced by:  ablfaclem3  20019