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

Theorem dprdsn 18258
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 2610 . . 3 (Cntz‘𝐺) = (Cntz‘𝐺)
2 eqid 2610 . . 3 (0g𝐺) = (0g𝐺)
3 eqid 2610 . . 3 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4 subgrcl 17422 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54adantl 481 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6 snex 4835 . . . 4 {𝐴} ∈ V
76a1i 11 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8 f1osng 6089 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9 f1of 6050 . . . . 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 476 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
1211snssd 4281 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
1310, 12fssd 5970 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14 simpr1 1060 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 ∈ {𝐴})
15 elsni 4142 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
1614, 15syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝐴)
17 simpr2 1061 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 ∈ {𝐴})
18 elsni 4142 . . . . . 6 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
1917, 18syl 17 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑦 = 𝐴)
2016, 19eqtr4d 2647 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥 = 𝑦)
21 simpr3 1062 . . . 4 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → 𝑥𝑦)
2220, 21pm2.21ddne 2866 . . 3 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥𝑦)) → ({⟨𝐴, 𝑆⟩}‘𝑥) ⊆ ((Cntz‘𝐺)‘({⟨𝐴, 𝑆⟩}‘𝑦)))
235adantr 480 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝐺 ∈ Grp)
24 eqid 2610 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
2524subgacs 17452 . . . . . . . 8 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26 acsmre 16136 . . . . . . . 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 481 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐴)
2928sneqd 4137 . . . . . . . . . . . . . 14 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
3029difeq2d 3690 . . . . . . . . . . . . 13 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31 difid 3902 . . . . . . . . . . . . 13 ({𝐴} ∖ {𝐴}) = ∅
3230, 31syl6eq 2660 . . . . . . . . . . . 12 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
3332imaeq2d 5385 . . . . . . . . . . 11 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34 ima0 5400 . . . . . . . . . . 11 ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
3533, 34syl6eq 2660 . . . . . . . . . 10 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
3635unieqd 4382 . . . . . . . . 9 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
37 uni0 4401 . . . . . . . . 9 ∅ = ∅
3836, 37syl6eq 2660 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39 0ss 3924 . . . . . . . . 9 ∅ ⊆ {(0g𝐺)}
4039a1i 11 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g𝐺)})
4138, 40eqsstrd 3602 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g𝐺)})
4220subg 17442 . . . . . . . 8 (𝐺 ∈ Grp → {(0g𝐺)} ∈ (SubGrp‘𝐺))
4323, 42syl 17 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ∈ (SubGrp‘𝐺))
443mrcsscl 16103 . . . . . . 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 1318 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g𝐺)})
462subg0cl 17425 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
4746ad2antlr 759 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ 𝑆)
4815fveq2d 6107 . . . . . . . . 9 (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49 fvsng 6352 . . . . . . . . 9 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
5048, 49sylan9eqr 2666 . . . . . . . 8 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
5147, 50eleqtrrd 2691 . . . . . . 7 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
5251snssd 4281 . . . . . 6 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
5345, 52sstrd 3578 . . . . 5 (((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54 sseqin2 3779 . . . . 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 207 . . . 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 3602 . . 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 18221 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
583dprdspan 18249 . . . 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 5532 . . . . . . . 8 (𝐴𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6160adantr 480 . . . . . . 7 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
6261unieqd 4382 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
63 unisng 4388 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
6463adantl 481 . . . . . 6 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} = 𝑆)
6562, 64eqtrd 2644 . . . . 5 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = 𝑆)
6665fveq2d 6107 . . . 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 16096 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
6967, 68sylancom 698 . . . 4 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
7066, 69eqtrd 2644 . . 3 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
7159, 70eqtrd 2644 . 2 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆)
7257, 71jca 553 1 ((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  cdif 3537  cin 3539  wss 3540  c0 3874  {csn 4125  cop 4131   cuni 4372   class class class wbr 4583  dom cdm 5038  ran crn 5039  cima 5041  wf 5800  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  Basecbs 15695  0gc0g 15923  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068  Grpcgrp 17245  SubGrpcsubg 17411  Cntzccntz 17571   DProd cdprd 18215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-gim 17524  df-cntz 17573  df-oppg 17599  df-cmn 18018  df-dprd 18217
This theorem is referenced by:  dprd2da  18264  dmdprdpr  18271  dprdpr  18272  dpjlem  18273  pgpfaclem1  18303
  Copyright terms: Public domain W3C validator