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Theorem dprdsn 19277
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 2738 . . 3 (Cntz‘𝐺) = (Cntz‘𝐺)
2 eqid 2738 . . 3 (0g𝐺) = (0g𝐺)
3 eqid 2738 . . 3 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4 subgrcl 18402 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54adantl 485 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6 snex 5298 . . . 4 {𝐴} ∈ V
76a1i 11 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8 f1osng 6658 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9 f1of 6618 . . . . 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 488 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
1211snssd 4697 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
1310, 12fssd 6522 . . 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 4533 . . . . . 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 4533 . . . . . 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 3018 . . 3 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → ({⟨𝐴, 𝑆⟩}‘𝑥) ⊆ ((Cntz‘𝐺)‘({⟨𝐴, 𝑆⟩}‘𝑦)))
235adantr 484 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝐺 ∈ Grp)
24 eqid 2738 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
2524subgacs 18431 . . . . . . . 8 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26 acsmre 17026 . . . . . . . 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 485 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐴)
2928sneqd 4528 . . . . . . . . . . . . . 14 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
3029difeq2d 4013 . . . . . . . . . . . . 13 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31 difid 4259 . . . . . . . . . . . . 13 ({𝐴} ∖ {𝐴}) = ∅
3230, 31eqtrdi 2789 . . . . . . . . . . . 12 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
3332imaeq2d 5903 . . . . . . . . . . 11 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34 ima0 5919 . . . . . . . . . . 11 ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
3533, 34eqtrdi 2789 . . . . . . . . . 10 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
3635unieqd 4810 . . . . . . . . 9 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
37 uni0 4826 . . . . . . . . 9 ∅ = ∅
3836, 37eqtrdi 2789 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39 0ss 4285 . . . . . . . . 9 ∅ ⊆ {(0g𝐺)}
4039a1i 11 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g𝐺)})
4138, 40eqsstrd 3915 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g𝐺)})
4220subg 18422 . . . . . . . 8 (𝐺 ∈ Grp → {(0g𝐺)} ∈ (SubGrp‘𝐺))
4323, 42syl 17 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ∈ (SubGrp‘𝐺))
443mrcsscl 16994 . . . . . . 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 18405 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
4746ad2antlr 727 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ 𝑆)
4815fveq2d 6678 . . . . . . . . 9 (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49 fvsng 6952 . . . . . . . . 9 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
5048, 49sylan9eqr 2795 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
5147, 50eleqtrrd 2836 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
5251snssd 4697 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
5345, 52sstrd 3887 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54 sseqin2 4106 . . . . 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 221 . . . 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 3915 . . 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 19240 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
583dprdspan 19268 . . . 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 6053 . . . . . . . 8 (𝐴𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6160adantr 484 . . . . . . 7 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6261unieqd 4810 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
63 unisng 4817 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
6463adantl 485 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} = 𝑆)
6562, 64eqtrd 2773 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = 𝑆)
6665fveq2d 6678 . . . 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 16987 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
6967, 68sylancom 591 . . . 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 515 1 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2934  Vcvv 3398  cdif 3840  cin 3842  wss 3843  c0 4211  {csn 4516  cop 4522   cuni 4796   class class class wbr 5030  dom cdm 5525  ran crn 5526  cima 5528  wf 6335  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7170  Basecbs 16586  0gc0g 16816  Moorecmre 16956  mrClscmrc 16957  ACScacs 16959  Grpcgrp 18219  SubGrpcsubg 18391  Cntzccntz 18563   DProd cdprd 19234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-supp 7857  df-tpos 7921  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-ixp 8508  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fsupp 8907  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-fzo 13125  df-seq 13461  df-hash 13783  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-0g 16818  df-gsum 16819  df-mre 16960  df-mrc 16961  df-acs 16963  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-mhm 18072  df-submnd 18073  df-grp 18222  df-minusg 18223  df-sbg 18224  df-mulg 18343  df-subg 18394  df-ghm 18474  df-gim 18517  df-cntz 18565  df-oppg 18592  df-cmn 19026  df-dprd 19236
This theorem is referenced by:  dprd2da  19283  dmdprdpr  19290  dprdpr  19291  dpjlem  19292  pgpfaclem1  19322
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