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Theorem frgpup3 18014
 Description: Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
frgpup3.g 𝐺 = (freeGrp‘𝐼)
frgpup3.b 𝐵 = (Base‘𝐻)
frgpup3.u 𝑈 = (varFGrp𝐼)
Assertion
Ref Expression
frgpup3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚𝑈) = 𝐹)
Distinct variable groups:   𝐵,𝑚   𝑚,𝐹   𝑚,𝐺   𝑚,𝐻   𝑚,𝐼   𝑈,𝑚   𝑚,𝑉

Proof of Theorem frgpup3
Dummy variables 𝑔 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup3.b . . 3 𝐵 = (Base‘𝐻)
2 eqid 2610 . . 3 (invg𝐻) = (invg𝐻)
3 eqid 2610 . . 3 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦))))
4 simp1 1054 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐻 ∈ Grp)
5 simp2 1055 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐼𝑉)
6 simp3 1056 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
7 eqid 2610 . . 3 ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
8 eqid 2610 . . 3 ( ~FG𝐼) = ( ~FG𝐼)
9 frgpup3.g . . 3 𝐺 = (freeGrp‘𝐼)
10 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2610 . . 3 ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) = ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11frgpup1 18011 . 2 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∈ (𝐺 GrpHom 𝐻))
134adantr 480 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝐻 ∈ Grp)
145adantr 480 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝐼𝑉)
156adantr 480 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝐹:𝐼𝐵)
16 frgpup3.u . . . . 5 𝑈 = (varFGrp𝐼)
17 simpr 476 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝑘𝐼)
181, 2, 3, 13, 14, 15, 7, 8, 9, 10, 11, 16, 17frgpup2 18012 . . . 4 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩)‘(𝑈𝑘)) = (𝐹𝑘))
1918mpteq2dva 4672 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → (𝑘𝐼 ↦ (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩)‘(𝑈𝑘))) = (𝑘𝐼 ↦ (𝐹𝑘)))
2010, 1ghmf 17487 . . . . 5 (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∈ (𝐺 GrpHom 𝐻) → ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩):(Base‘𝐺)⟶𝐵)
2112, 20syl 17 . . . 4 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩):(Base‘𝐺)⟶𝐵)
228, 16, 9, 10vrgpf 18004 . . . . 5 (𝐼𝑉𝑈:𝐼⟶(Base‘𝐺))
235, 22syl 17 . . . 4 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝑈:𝐼⟶(Base‘𝐺))
24 fcompt 6306 . . . 4 ((ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩):(Base‘𝐺)⟶𝐵𝑈:𝐼⟶(Base‘𝐺)) → (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∘ 𝑈) = (𝑘𝐼 ↦ (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩)‘(𝑈𝑘))))
2521, 23, 24syl2anc 691 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∘ 𝑈) = (𝑘𝐼 ↦ (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩)‘(𝑈𝑘))))
266feqmptd 6159 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐹 = (𝑘𝐼 ↦ (𝐹𝑘)))
2719, 25, 263eqtr4d 2654 . 2 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∘ 𝑈) = 𝐹)
284adantr 480 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝐻 ∈ Grp)
295adantr 480 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝐼𝑉)
306adantr 480 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝐹:𝐼𝐵)
31 simprl 790 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝑚 ∈ (𝐺 GrpHom 𝐻))
32 simprr 792 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → (𝑚𝑈) = 𝐹)
331, 2, 3, 28, 29, 30, 7, 8, 9, 10, 11, 16, 31, 32frgpup3lem 18013 . . . 4 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝑚 = ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩))
3433expr 641 . . 3 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑚 ∈ (𝐺 GrpHom 𝐻)) → ((𝑚𝑈) = 𝐹𝑚 = ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩)))
3534ralrimiva 2949 . 2 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ∀𝑚 ∈ (𝐺 GrpHom 𝐻)((𝑚𝑈) = 𝐹𝑚 = ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩)))
36 coeq1 5201 . . . 4 (𝑚 = ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) → (𝑚𝑈) = (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∘ 𝑈))
3736eqeq1d 2612 . . 3 (𝑚 = ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) → ((𝑚𝑈) = 𝐹 ↔ (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∘ 𝑈) = 𝐹))
3837eqreu 3365 . 2 ((ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∈ (𝐺 GrpHom 𝐻) ∧ (ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩) ∘ 𝑈) = 𝐹 ∧ ∀𝑚 ∈ (𝐺 GrpHom 𝐻)((𝑚𝑈) = 𝐹𝑚 = ran (𝑔 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ ⟨[𝑔]( ~FG𝐼), (𝐻 Σg ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) ∘ 𝑔))⟩))) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚𝑈) = 𝐹)
3912, 27, 35, 38syl3anc 1318 1 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚𝑈) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898  ∅c0 3874  ifcif 4036  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  2𝑜c2o 7441  [cec 7627  Word cword 13146  Basecbs 15695   Σg cgsu 15924  Grpcgrp 17245  invgcminusg 17246   GrpHom cghm 17480   ~FG cefg 17942  freeGrpcfrgp 17943  varFGrpcvrgp 17944 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-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-ot 4134  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-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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  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-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-0g 15925  df-gsum 15926  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-frmd 17209  df-vrmd 17210  df-grp 17248  df-minusg 17249  df-ghm 17481  df-efg 17945  df-frgp 17946  df-vrgp 17947 This theorem is referenced by:  0frgp  18015  frgpcyg  19741
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