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Theorem dprdsn 19157
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 2821 . . 3 (Cntz‘𝐺) = (Cntz‘𝐺)
2 eqid 2821 . . 3 (0g𝐺) = (0g𝐺)
3 eqid 2821 . . 3 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4 subgrcl 18283 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54adantl 484 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6 snex 5331 . . . 4 {𝐴} ∈ V
76a1i 11 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8 f1osng 6654 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9 f1of 6614 . . . . 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 487 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
1211snssd 4741 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
1310, 12fssd 6527 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14 simpr1 1190 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 ∈ {𝐴})
15 elsni 4583 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
1614, 15syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝐴)
17 simpr2 1191 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 ∈ {𝐴})
18 elsni 4583 . . . . . 6 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
1917, 18syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 = 𝐴)
2016, 19eqtr4d 2859 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝑦)
21 simpr3 1192 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥𝑦)
2220, 21pm2.21ddne 3101 . . 3 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → ({⟨𝐴, 𝑆⟩}‘𝑥) ⊆ ((Cntz‘𝐺)‘({⟨𝐴, 𝑆⟩}‘𝑦)))
235adantr 483 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝐺 ∈ Grp)
24 eqid 2821 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
2524subgacs 18312 . . . . . . . 8 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26 acsmre 16922 . . . . . . . 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 484 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐴)
2928sneqd 4578 . . . . . . . . . . . . . 14 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
3029difeq2d 4098 . . . . . . . . . . . . 13 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31 difid 4329 . . . . . . . . . . . . 13 ({𝐴} ∖ {𝐴}) = ∅
3230, 31syl6eq 2872 . . . . . . . . . . . 12 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
3332imaeq2d 5928 . . . . . . . . . . 11 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34 ima0 5944 . . . . . . . . . . 11 ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
3533, 34syl6eq 2872 . . . . . . . . . 10 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
3635unieqd 4851 . . . . . . . . 9 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
37 uni0 4865 . . . . . . . . 9 ∅ = ∅
3836, 37syl6eq 2872 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39 0ss 4349 . . . . . . . . 9 ∅ ⊆ {(0g𝐺)}
4039a1i 11 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g𝐺)})
4138, 40eqsstrd 4004 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g𝐺)})
4220subg 18303 . . . . . . . 8 (𝐺 ∈ Grp → {(0g𝐺)} ∈ (SubGrp‘𝐺))
4323, 42syl 17 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ∈ (SubGrp‘𝐺))
443mrcsscl 16890 . . . . . . 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 1367 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g𝐺)})
462subg0cl 18286 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
4746ad2antlr 725 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ 𝑆)
4815fveq2d 6673 . . . . . . . . 9 (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49 fvsng 6941 . . . . . . . . 9 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
5048, 49sylan9eqr 2878 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
5147, 50eleqtrrd 2916 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
5251snssd 4741 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
5345, 52sstrd 3976 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54 sseqin2 4191 . . . . 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 220 . . . 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 4004 . . 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 19120 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
583dprdspan 19148 . . . 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 6077 . . . . . . . 8 (𝐴𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6160adantr 483 . . . . . . 7 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6261unieqd 4851 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
63 unisng 4856 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
6463adantl 484 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} = 𝑆)
6562, 64eqtrd 2856 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = 𝑆)
6665fveq2d 6673 . . . 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 16883 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
6967, 68sylancom 590 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
7066, 69eqtrd 2856 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
7159, 70eqtrd 2856 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆)
7257, 71jca 514 1 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  Vcvv 3494  cdif 3932  cin 3934  wss 3935  c0 4290  {csn 4566  cop 4572   cuni 4837   class class class wbr 5065  dom cdm 5554  ran crn 5555  cima 5557  wf 6350  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7155  Basecbs 16482  0gc0g 16712  Moorecmre 16852  mrClscmrc 16853  ACScacs 16855  Grpcgrp 18102  SubGrpcsubg 18272  Cntzccntz 18444   DProd cdprd 19114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-supp 7830  df-tpos 7891  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-ixp 8461  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-fsupp 8833  df-oi 8973  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-seq 13369  df-hash 13690  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-0g 16714  df-gsum 16715  df-mre 16856  df-mrc 16857  df-acs 16859  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-mhm 17955  df-submnd 17956  df-grp 18105  df-minusg 18106  df-sbg 18107  df-mulg 18224  df-subg 18275  df-ghm 18355  df-gim 18398  df-cntz 18446  df-oppg 18473  df-cmn 18907  df-dprd 19116
This theorem is referenced by:  dprd2da  19163  dmdprdpr  19170  dprdpr  19171  dpjlem  19172  pgpfaclem1  19202
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