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Theorem subgdprd 19157
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 18282 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3syl 17 . . . . 5 (𝜑𝐻 ∈ Grp)
5 eqid 2824 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
65subgacs 18313 . . . . 5 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7 acsmre 16923 . . . . 5 ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
84, 6, 73syl 18 . . . 4 (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
9 subgrcl 18284 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
101, 9syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
11 eqid 2824 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1211subgacs 18313 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13 acsmre 16923 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
1410, 12, 133syl 18 . . . . 5 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
15 eqid 2824 . . . . 5 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16 subgdprd.3 . . . . . . . 8 (𝜑𝐺dom DProd 𝑆)
17 dprdf 19128 . . . . . . . 8 (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18 frn 6509 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
1916, 17, 183syl 18 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20 mresspw 16863 . . . . . . . 8 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2114, 20syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2219, 21sstrd 3963 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23 sspwuni 5008 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
2422, 23sylib 221 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
2514, 15, 24mrcssidd 16896 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
2615mrccl 16882 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
2714, 24, 26syl2anc 587 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
28 subgdprd.4 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29 sspwuni 5008 . . . . . . 7 (ran 𝑆 ⊆ 𝒫 𝐴 ran 𝑆𝐴)
3028, 29sylib 221 . . . . . 6 (𝜑 ran 𝑆𝐴)
3115mrcsscl 16891 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆𝐴𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
3214, 30, 1, 31syl3anc 1368 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
332subsubg 18302 . . . . . 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 712 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
36 eqid 2824 . . . . 5 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
3736mrcsscl 16891 . . . 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 1368 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
392subgdmdprd 19156 . . . . . . . . . . 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 712 . . . . . . . . 9 (𝜑𝐻dom DProd 𝑆)
42 eqidd 2825 . . . . . . . . 9 (𝜑 → dom 𝑆 = dom 𝑆)
4341, 42dprdf2 19129 . . . . . . . 8 (𝜑𝑆:dom 𝑆⟶(SubGrp‘𝐻))
4443frnd 6510 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45 mresspw 16863 . . . . . . . 8 ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
468, 45syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
4744, 46sstrd 3963 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48 sspwuni 5008 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ran 𝑆 ⊆ (Base‘𝐻))
4947, 48sylib 221 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐻))
508, 36, 49mrcssidd 16896 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5136mrccl 16882 . . . . . . 7 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
528, 49, 51syl2anc 587 . . . . . 6 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
532subsubg 18302 . . . . . . 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 235 . . . . 5 (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴))
5655simpld 498 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
5715mrcsscl 16891 . . . 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 1368 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5938, 58eqssd 3970 . 2 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6036dprdspan 19149 . . 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 19149 . . 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 2869 1 (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wss 3919  𝒫 cpw 4522   cuni 4824   class class class wbr 5052  dom cdm 5542  ran crn 5543  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483  s cress 16484  Moorecmre 16853  mrClscmrc 16854  ACScacs 16856  Grpcgrp 18103  SubGrpcsubg 18273   DProd cdprd 19115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-tpos 7888  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-0g 16715  df-gsum 16716  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-gim 18399  df-cntz 18447  df-oppg 18474  df-cmn 18908  df-dprd 19117
This theorem is referenced by:  ablfaclem3  19209
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