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Theorem subgdprd 19957
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 19056 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3syl 17 . . . . 5 (𝜑𝐻 ∈ Grp)
5 eqid 2726 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
65subgacs 19088 . . . . 5 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
7 acsmre 17605 . . . . 5 ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
84, 6, 73syl 18 . . . 4 (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
9 subgrcl 19058 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
101, 9syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
11 eqid 2726 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1211subgacs 19088 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
13 acsmre 17605 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
1410, 12, 133syl 18 . . . . 5 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
15 eqid 2726 . . . . 5 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
16 subgdprd.3 . . . . . . . 8 (𝜑𝐺dom DProd 𝑆)
17 dprdf 19928 . . . . . . . 8 (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
18 frn 6718 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
1916, 17, 183syl 18 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
20 mresspw 17545 . . . . . . . 8 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2114, 20syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
2219, 21sstrd 3987 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
23 sspwuni 5096 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
2422, 23sylib 217 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
2514, 15, 24mrcssidd 17578 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
2615mrccl 17564 . . . . . 6 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran 𝑆 ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
2714, 24, 26syl2anc 583 . . . . 5 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐺))
28 subgdprd.4 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)
29 sspwuni 5096 . . . . . . 7 (ran 𝑆 ⊆ 𝒫 𝐴 ran 𝑆𝐴)
3028, 29sylib 217 . . . . . 6 (𝜑 ran 𝑆𝐴)
3115mrcsscl 17573 . . . . . 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 19076 . . . . . 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 710 . . . 4 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
36 eqid 2726 . . . . 5 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
3736mrcsscl 17573 . . . 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 19956 . . . . . . . . . . 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 710 . . . . . . . . 9 (𝜑𝐻dom DProd 𝑆)
42 eqidd 2727 . . . . . . . . 9 (𝜑 → dom 𝑆 = dom 𝑆)
4341, 42dprdf2 19929 . . . . . . . 8 (𝜑𝑆:dom 𝑆⟶(SubGrp‘𝐻))
4443frnd 6719 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐻))
45 mresspw 17545 . . . . . . . 8 ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
468, 45syl 17 . . . . . . 7 (𝜑 → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
4744, 46sstrd 3987 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐻))
48 sspwuni 5096 . . . . . 6 (ran 𝑆 ⊆ 𝒫 (Base‘𝐻) ↔ ran 𝑆 ⊆ (Base‘𝐻))
4947, 48sylib 217 . . . . 5 (𝜑 ran 𝑆 ⊆ (Base‘𝐻))
508, 36, 49mrcssidd 17578 . . . 4 (𝜑 ran 𝑆 ⊆ ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆))
5136mrccl 17564 . . . . . . 7 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ran 𝑆 ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
528, 49, 51syl2anc 583 . . . . . 6 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) ∈ (SubGrp‘𝐻))
532subsubg 19076 . . . . . . 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 231 . . . . 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 17573 . . . 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 3994 . 2 (𝜑 → ((mrCls‘(SubGrp‘𝐻))‘ ran 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
6036dprdspan 19949 . . 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 19949 . . 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 2776 1 (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wss 3943  𝒫 cpw 4597   cuni 4902   class class class wbr 5141  dom cdm 5669  ran crn 5670  wf 6533  cfv 6537  (class class class)co 7405  Basecbs 17153  s cress 17182  Moorecmre 17535  mrClscmrc 17536  ACScacs 17538  Grpcgrp 18863  SubGrpcsubg 19047   DProd cdprd 19915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-gsum 17397  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18996  df-subg 19050  df-ghm 19139  df-gim 19184  df-cntz 19233  df-oppg 19262  df-cmn 19702  df-dprd 19917
This theorem is referenced by:  ablfaclem3  20009
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