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Theorem subgdprd 20055
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 19147 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3syl 17 . . . . 5 (𝜑𝐻 ∈ Grp)
5 eqid 2737 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
65subgacs 19179 . . . . 5 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7 acsmre 17695 . . . . 5 ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
84, 6, 73syl 18 . . . 4 (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
9 subgrcl 19149 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
101, 9syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
11 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1211subgacs 19179 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13 acsmre 17695 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
1410, 12, 133syl 18 . . . . 5 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
15 eqid 2737 . . . . 5 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16 subgdprd.3 . . . . . . . 8 (𝜑𝐺dom DProd 𝑆)
17 dprdf 20026 . . . . . . . 8 (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18 frn 6743 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
1916, 17, 183syl 18 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20 mresspw 17635 . . . . . . . 8 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2114, 20syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2219, 21sstrd 3994 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23 sspwuni 5100 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
2422, 23sylib 218 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
2514, 15, 24mrcssidd 17668 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
2615mrccl 17654 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
2714, 24, 26syl2anc 584 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
28 subgdprd.4 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29 sspwuni 5100 . . . . . . 7 (ran 𝑆 ⊆ 𝒫 𝐴 ran 𝑆𝐴)
3028, 29sylib 218 . . . . . 6 (𝜑 ran 𝑆𝐴)
3115mrcsscl 17663 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆𝐴𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
3214, 30, 1, 31syl3anc 1373 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ 𝐴)
332subsubg 19167 . . . . . 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 713 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
36 eqid 2737 . . . . 5 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
3736mrcsscl 17663 . . . 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 1373 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
392subgdmdprd 20054 . . . . . . . . . . 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 713 . . . . . . . . 9 (𝜑𝐻dom DProd 𝑆)
42 eqidd 2738 . . . . . . . . 9 (𝜑 → dom 𝑆 = dom 𝑆)
4341, 42dprdf2 20027 . . . . . . . 8 (𝜑𝑆:dom 𝑆⟶(SubGrp‘𝐻))
4443frnd 6744 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45 mresspw 17635 . . . . . . . 8 ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
468, 45syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
4744, 46sstrd 3994 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48 sspwuni 5100 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ran 𝑆 ⊆ (Base‘𝐻))
4947, 48sylib 218 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐻))
508, 36, 49mrcssidd 17668 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5136mrccl 17654 . . . . . . 7 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
528, 49, 51syl2anc 584 . . . . . 6 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
532subsubg 19167 . . . . . . 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 232 . . . . 5 (𝜑 → (((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ⊆ 𝐴))
5655simpld 494 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
5715mrcsscl 17663 . . . 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 1373 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5938, 58eqssd 4001 . 2 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6036dprdspan 20047 . . 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 20047 . . 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 2787 1 (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Moorecmre 17625  mrClscmrc 17626  ACScacs 17628  Grpcgrp 18951  SubGrpcsubg 19138   DProd cdprd 20013
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-gim 19277  df-cntz 19335  df-oppg 19364  df-cmn 19800  df-dprd 20015
This theorem is referenced by:  ablfaclem3  20107
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