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Theorem subgdprd 18630
Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypotheses
Ref Expression
subgdprd.1 𝐻 = (𝐺s 𝐴)
subgdprd.2 (𝜑𝐴 ∈ (SubGrp‘𝐺))
subgdprd.3 (𝜑𝐺dom DProd 𝑆)
subgdprd.4 (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
Assertion
Ref Expression
subgdprd (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

Proof of Theorem subgdprd
StepHypRef Expression
1 subgdprd.2 . . . . . 6 (𝜑𝐴 ∈ (SubGrp‘𝐺))
2 subgdprd.1 . . . . . . 7 𝐻 = (𝐺s 𝐴)
32subggrp 17794 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3syl 17 . . . . 5 (𝜑𝐻 ∈ Grp)
5 eqid 2756 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
65subgacs 17826 . . . . 5 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7 acsmre 16510 . . . . 5 ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
84, 6, 73syl 18 . . . 4 (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
9 subgrcl 17796 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
101, 9syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
11 eqid 2756 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1211subgacs 17826 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13 acsmre 16510 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
1410, 12, 133syl 18 . . . . 5 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
15 eqid 2756 . . . . 5 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16 subgdprd.3 . . . . . . . 8 (𝜑𝐺dom DProd 𝑆)
17 dprdf 18601 . . . . . . . 8 (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18 frn 6210 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
1916, 17, 183syl 18 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20 mresspw 16450 . . . . . . . 8 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2114, 20syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2219, 21sstrd 3750 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23 sspwuni 4759 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
2422, 23sylib 208 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
2514, 15, 24mrcssidd 16483 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
2615mrccl 16469 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
2714, 24, 26syl2anc 696 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
28 subgdprd.4 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29 sspwuni 4759 . . . . . . 7 (ran 𝑆 ⊆ 𝒫 𝐴 ran 𝑆𝐴)
3028, 29sylib 208 . . . . . 6 (𝜑 ran 𝑆𝐴)
3115mrcsscl 16478 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆𝐴𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
3214, 30, 1, 31syl3anc 1477 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
332subsubg 17814 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → (((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)))
341, 33syl 17 . . . . 5 (𝜑 → (((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)))
3527, 32, 34mpbir2and 995 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
36 eqid 2756 . . . . 5 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
3736mrcsscl 16478 . . . 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))
388, 25, 35, 37syl3anc 1477 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
392subgdmdprd 18629 . . . . . . . . . . 11 (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
401, 39syl 17 . . . . . . . . . 10 (𝜑 → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
4116, 28, 40mpbir2and 995 . . . . . . . . 9 (𝜑𝐻dom DProd 𝑆)
42 eqidd 2757 . . . . . . . . 9 (𝜑 → dom 𝑆 = dom 𝑆)
4341, 42dprdf2 18602 . . . . . . . 8 (𝜑𝑆:dom 𝑆⟶(SubGrp‘𝐻))
44 frn 6210 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐻) → ran 𝑆 ⊆ (SubGrp‘𝐻))
4543, 44syl 17 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
46 mresspw 16450 . . . . . . . 8 ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
478, 46syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
4845, 47sstrd 3750 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
49 sspwuni 4759 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ran 𝑆 ⊆ (Base‘𝐻))
5048, 49sylib 208 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐻))
518, 36, 50mrcssidd 16483 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5236mrccl 16469 . . . . . . 7 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
538, 50, 52syl2anc 696 . . . . . 6 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
542subsubg 17814 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴)))
551, 54syl 17 . . . . . 6 (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴)))
5653, 55mpbid 222 . . . . 5 (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴))
5756simpld 477 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
5815mrcsscl 16478 . . . 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))
5914, 51, 57, 58syl3anc 1477 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
6038, 59eqssd 3757 . 2 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6136dprdspan 18622 . . 3 (𝐻dom DProd 𝑆 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
6241, 61syl 17 . 2 (𝜑 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
6315dprdspan 18622 . . 3 (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6416, 63syl 17 . 2 (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6560, 62, 643eqtr4d 2800 1 (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1628  wcel 2135  wss 3711  𝒫 cpw 4298   cuni 4584   class class class wbr 4800  dom cdm 5262  ran crn 5263  wf 6041  cfv 6045  (class class class)co 6809  Basecbs 16055  s cress 16056  Moorecmre 16440  mrClscmrc 16441  ACScacs 16443  Grpcgrp 17619  SubGrpcsubg 17785   DProd cdprd 18588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-inf2 8707  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-iin 4671  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-se 5222  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-isom 6054  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-of 7058  df-om 7227  df-1st 7329  df-2nd 7330  df-supp 7460  df-tpos 7517  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-map 8021  df-ixp 8071  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-fsupp 8437  df-oi 8576  df-card 8951  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-2 11267  df-n0 11481  df-z 11566  df-uz 11876  df-fz 12516  df-fzo 12656  df-seq 12992  df-hash 13308  df-ndx 16058  df-slot 16059  df-base 16061  df-sets 16062  df-ress 16063  df-plusg 16152  df-0g 16300  df-gsum 16301  df-mre 16444  df-mrc 16445  df-acs 16447  df-mgm 17439  df-sgrp 17481  df-mnd 17492  df-mhm 17532  df-submnd 17533  df-grp 17622  df-minusg 17623  df-sbg 17624  df-mulg 17738  df-subg 17788  df-ghm 17855  df-gim 17898  df-cntz 17946  df-oppg 17972  df-cmn 18391  df-dprd 18590
This theorem is referenced by:  ablfaclem3  18682
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