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Theorem dprdsn 19897
Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
dprdsn ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))

Proof of Theorem dprdsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Cntz‘𝐺) = (Cntz‘𝐺)
2 eqid 2733 . . 3 (0g𝐺) = (0g𝐺)
3 eqid 2733 . . 3 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4 subgrcl 19004 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54adantl 483 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6 snex 5429 . . . 4 {𝐴} ∈ V
76a1i 11 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8 f1osng 6870 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9 f1of 6829 . . . . 5 ({⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆} → {⟨𝐴, 𝑆⟩}:{𝐴}⟶{𝑆})
108, 9syl 17 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶{𝑆})
11 simpr 486 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
1211snssd 4810 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
1310, 12fssd 6731 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14 simpr1 1195 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 ∈ {𝐴})
15 elsni 4643 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
1614, 15syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝐴)
17 simpr2 1196 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 ∈ {𝐴})
18 elsni 4643 . . . . . 6 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
1917, 18syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 = 𝐴)
2016, 19eqtr4d 2776 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝑦)
21 simpr3 1197 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥𝑦)
2220, 21pm2.21ddne 3027 . . 3 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → ({⟨𝐴, 𝑆⟩}‘𝑥) ⊆ ((Cntz‘𝐺)‘({⟨𝐴, 𝑆⟩}‘𝑦)))
235adantr 482 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝐺 ∈ Grp)
24 eqid 2733 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
2524subgacs 19034 . . . . . . . 8 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26 acsmre 17591 . . . . . . . 8 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
2723, 25, 263syl 18 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
2815adantl 483 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐴)
2928sneqd 4638 . . . . . . . . . . . . . 14 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
3029difeq2d 4120 . . . . . . . . . . . . 13 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31 difid 4368 . . . . . . . . . . . . 13 ({𝐴} ∖ {𝐴}) = ∅
3230, 31eqtrdi 2789 . . . . . . . . . . . 12 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
3332imaeq2d 6056 . . . . . . . . . . 11 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34 ima0 6072 . . . . . . . . . . 11 ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
3533, 34eqtrdi 2789 . . . . . . . . . 10 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
3635unieqd 4920 . . . . . . . . 9 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
37 uni0 4937 . . . . . . . . 9 ∅ = ∅
3836, 37eqtrdi 2789 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39 0ss 4394 . . . . . . . . 9 ∅ ⊆ {(0g𝐺)}
4039a1i 11 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g𝐺)})
4138, 40eqsstrd 4018 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g𝐺)})
4220subg 19024 . . . . . . . 8 (𝐺 ∈ Grp → {(0g𝐺)} ∈ (SubGrp‘𝐺))
4323, 42syl 17 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ∈ (SubGrp‘𝐺))
443mrcsscl 17559 . . . . . . 7 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g𝐺)} ∧ {(0g𝐺)} ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g𝐺)})
4527, 41, 43, 44syl3anc 1372 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g𝐺)})
462subg0cl 19007 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
4746ad2antlr 726 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ 𝑆)
4815fveq2d 6891 . . . . . . . . 9 (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49 fvsng 7172 . . . . . . . . 9 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
5048, 49sylan9eqr 2795 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
5147, 50eleqtrrd 2837 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
5251snssd 4810 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
5345, 52sstrd 3990 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54 sseqin2 4213 . . . . 5 (((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥) ↔ (({⟨𝐴, 𝑆⟩}‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})))) = ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))))
5553, 54sylib 217 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (({⟨𝐴, 𝑆⟩}‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})))) = ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))))
5655, 45eqsstrd 4018 . . 3 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (({⟨𝐴, 𝑆⟩}‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})))) ⊆ {(0g𝐺)})
571, 2, 3, 5, 7, 13, 22, 56dmdprdd 19860 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
583dprdspan 19888 . . . 4 (𝐺dom DProd {⟨𝐴, 𝑆⟩} → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = ((mrCls‘(SubGrp‘𝐺))‘ ran {⟨𝐴, 𝑆⟩}))
5957, 58syl 17 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = ((mrCls‘(SubGrp‘𝐺))‘ ran {⟨𝐴, 𝑆⟩}))
60 rnsnopg 6216 . . . . . . . 8 (𝐴𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6160adantr 482 . . . . . . 7 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6261unieqd 4920 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
63 unisng 4927 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
6463adantl 483 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} = 𝑆)
6562, 64eqtrd 2773 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = 𝑆)
6665fveq2d 6891 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran {⟨𝐴, 𝑆⟩}) = ((mrCls‘(SubGrp‘𝐺))‘𝑆))
675, 25, 263syl 18 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
683mrcid 17552 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
6967, 68sylancom 589 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
7066, 69eqtrd 2773 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
7159, 70eqtrd 2773 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆)
7257, 71jca 513 1 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cdif 3943  cin 3945  wss 3946  c0 4320  {csn 4626  cop 4632   cuni 4906   class class class wbr 5146  dom cdm 5674  ran crn 5675  cima 5677  wf 6535  1-1-ontowf1o 6538  cfv 6539  (class class class)co 7403  Basecbs 17139  0gc0g 17380  Moorecmre 17521  mrClscmrc 17522  ACScacs 17524  Grpcgrp 18814  SubGrpcsubg 18993  Cntzccntz 19172   DProd cdprd 19854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-iin 4998  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7664  df-om 7850  df-1st 7969  df-2nd 7970  df-supp 8141  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-1o 8460  df-er 8698  df-map 8817  df-ixp 8887  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fsupp 9357  df-oi 9500  df-card 9929  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-nn 12208  df-2 12270  df-n0 12468  df-z 12554  df-uz 12818  df-fz 13480  df-fzo 13623  df-seq 13962  df-hash 14286  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17140  df-ress 17169  df-plusg 17205  df-0g 17382  df-gsum 17383  df-mre 17525  df-mrc 17526  df-acs 17528  df-mgm 18556  df-sgrp 18605  df-mnd 18621  df-mhm 18666  df-submnd 18667  df-grp 18817  df-minusg 18818  df-sbg 18819  df-mulg 18944  df-subg 18996  df-ghm 19083  df-gim 19126  df-cntz 19174  df-oppg 19202  df-cmn 19642  df-dprd 19856
This theorem is referenced by:  dprd2da  19903  dmdprdpr  19910  dprdpr  19911  dpjlem  19912  pgpfaclem1  19942
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