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Theorem subgdprd 18897
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 18056 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3syl 17 . . . . 5 (𝜑𝐻 ∈ Grp)
5 eqid 2772 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
65subgacs 18088 . . . . 5 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7 acsmre 16771 . . . . 5 ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
84, 6, 73syl 18 . . . 4 (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
9 subgrcl 18058 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
101, 9syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
11 eqid 2772 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1211subgacs 18088 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13 acsmre 16771 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
1410, 12, 133syl 18 . . . . 5 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
15 eqid 2772 . . . . 5 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16 subgdprd.3 . . . . . . . 8 (𝜑𝐺dom DProd 𝑆)
17 dprdf 18868 . . . . . . . 8 (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18 frn 6344 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
1916, 17, 183syl 18 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20 mresspw 16711 . . . . . . . 8 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2114, 20syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2219, 21sstrd 3864 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23 sspwuni 4882 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
2422, 23sylib 210 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
2514, 15, 24mrcssidd 16744 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
2615mrccl 16730 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
2714, 24, 26syl2anc 576 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
28 subgdprd.4 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29 sspwuni 4882 . . . . . . 7 (ran 𝑆 ⊆ 𝒫 𝐴 ran 𝑆𝐴)
3028, 29sylib 210 . . . . . 6 (𝜑 ran 𝑆𝐴)
3115mrcsscl 16739 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆𝐴𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
3214, 30, 1, 31syl3anc 1351 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
332subsubg 18076 . . . . . 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 700 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
36 eqid 2772 . . . . 5 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
3736mrcsscl 16739 . . . 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 1351 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
392subgdmdprd 18896 . . . . . . . . . . 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 700 . . . . . . . . 9 (𝜑𝐻dom DProd 𝑆)
42 eqidd 2773 . . . . . . . . 9 (𝜑 → dom 𝑆 = dom 𝑆)
4341, 42dprdf2 18869 . . . . . . . 8 (𝜑𝑆:dom 𝑆⟶(SubGrp‘𝐻))
4443frnd 6345 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45 mresspw 16711 . . . . . . . 8 ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
468, 45syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
4744, 46sstrd 3864 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48 sspwuni 4882 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ran 𝑆 ⊆ (Base‘𝐻))
4947, 48sylib 210 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐻))
508, 36, 49mrcssidd 16744 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5136mrccl 16730 . . . . . . 7 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
528, 49, 51syl2anc 576 . . . . . 6 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
532subsubg 18076 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴)))
541, 53syl 17 . . . . . 6 (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴)))
5552, 54mpbid 224 . . . . 5 (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴))
5655simpld 487 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
5715mrcsscl 16739 . . . 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))
5814, 50, 56, 57syl3anc 1351 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5938, 58eqssd 3871 . 2 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6036dprdspan 18889 . . 3 (𝐻dom DProd 𝑆 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
6141, 60syl 17 . 2 (𝜑 → (𝐻 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
6215dprdspan 18889 . . 3 (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6316, 62syl 17 . 2 (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6459, 61, 633eqtr4d 2818 1 (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wss 3825  𝒫 cpw 4416   cuni 4706   class class class wbr 4923  dom cdm 5400  ran crn 5401  wf 6178  cfv 6182  (class class class)co 6970  Basecbs 16329  s cress 16330  Moorecmre 16701  mrClscmrc 16702  ACScacs 16704  Grpcgrp 17881  SubGrpcsubg 18047   DProd cdprd 18855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7494  df-2nd 7495  df-supp 7627  df-tpos 7688  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-map 8200  df-ixp 8252  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fsupp 8621  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-n0 11701  df-z 11787  df-uz 12052  df-fz 12702  df-fzo 12843  df-seq 13178  df-hash 13499  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-0g 16561  df-gsum 16562  df-mre 16705  df-mrc 16706  df-acs 16708  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-mhm 17793  df-submnd 17794  df-grp 17884  df-minusg 17885  df-sbg 17886  df-mulg 18002  df-subg 18050  df-ghm 18117  df-gim 18160  df-cntz 18208  df-oppg 18235  df-cmn 18658  df-dprd 18857
This theorem is referenced by:  ablfaclem3  18949
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