MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdsn Structured version   Visualization version   GIF version

Theorem dprdsn 19639
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 18760 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54adantl 482 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6 snex 5354 . . . 4 {𝐴} ∈ V
76a1i 11 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8 f1osng 6757 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9 f1of 6716 . . . . 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 485 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
1211snssd 4742 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
1310, 12fssd 6618 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14 simpr1 1193 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 ∈ {𝐴})
15 elsni 4578 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
1614, 15syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝐴)
17 simpr2 1194 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 ∈ {𝐴})
18 elsni 4578 . . . . . 6 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
1917, 18syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 = 𝐴)
2016, 19eqtr4d 2781 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝑦)
21 simpr3 1195 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥𝑦)
2220, 21pm2.21ddne 3029 . . 3 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → ({⟨𝐴, 𝑆⟩}‘𝑥) ⊆ ((Cntz‘𝐺)‘({⟨𝐴, 𝑆⟩}‘𝑦)))
235adantr 481 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝐺 ∈ Grp)
24 eqid 2738 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
2524subgacs 18789 . . . . . . . 8 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26 acsmre 17361 . . . . . . . 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 482 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐴)
2928sneqd 4573 . . . . . . . . . . . . . 14 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
3029difeq2d 4057 . . . . . . . . . . . . 13 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31 difid 4304 . . . . . . . . . . . . 13 ({𝐴} ∖ {𝐴}) = ∅
3230, 31eqtrdi 2794 . . . . . . . . . . . 12 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
3332imaeq2d 5969 . . . . . . . . . . 11 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34 ima0 5985 . . . . . . . . . . 11 ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
3533, 34eqtrdi 2794 . . . . . . . . . 10 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
3635unieqd 4853 . . . . . . . . 9 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
37 uni0 4869 . . . . . . . . 9 ∅ = ∅
3836, 37eqtrdi 2794 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39 0ss 4330 . . . . . . . . 9 ∅ ⊆ {(0g𝐺)}
4039a1i 11 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g𝐺)})
4138, 40eqsstrd 3959 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g𝐺)})
4220subg 18780 . . . . . . . 8 (𝐺 ∈ Grp → {(0g𝐺)} ∈ (SubGrp‘𝐺))
4323, 42syl 17 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ∈ (SubGrp‘𝐺))
443mrcsscl 17329 . . . . . . 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 1370 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g𝐺)})
462subg0cl 18763 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
4746ad2antlr 724 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ 𝑆)
4815fveq2d 6778 . . . . . . . . 9 (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49 fvsng 7052 . . . . . . . . 9 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
5048, 49sylan9eqr 2800 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
5147, 50eleqtrrd 2842 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
5251snssd 4742 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
5345, 52sstrd 3931 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54 sseqin2 4149 . . . . 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 3959 . . 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 19602 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
583dprdspan 19630 . . . 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 6124 . . . . . . . 8 (𝐴𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6160adantr 481 . . . . . . 7 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6261unieqd 4853 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
63 unisng 4860 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
6463adantl 482 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} = 𝑆)
6562, 64eqtrd 2778 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = 𝑆)
6665fveq2d 6778 . . . 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 17322 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
6967, 68sylancom 588 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
7066, 69eqtrd 2778 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
7159, 70eqtrd 2778 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆)
7257, 71jca 512 1 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  Vcvv 3432  cdif 3884  cin 3886  wss 3887  c0 4256  {csn 4561  cop 4567   cuni 4839   class class class wbr 5074  dom cdm 5589  ran crn 5590  cima 5592  wf 6429  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275  Basecbs 16912  0gc0g 17150  Moorecmre 17291  mrClscmrc 17292  ACScacs 17294  Grpcgrp 18577  SubGrpcsubg 18749  Cntzccntz 18921   DProd cdprd 19596
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-gim 18875  df-cntz 18923  df-oppg 18950  df-cmn 19388  df-dprd 19598
This theorem is referenced by:  dprd2da  19645  dmdprdpr  19652  dprdpr  19653  dpjlem  19654  pgpfaclem1  19684
  Copyright terms: Public domain W3C validator