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Theorem frgpup3 18112
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 2621 . . 3 (invg𝐻) = (invg𝐻)
3 eqid 2621 . . 3 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦)))) = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), ((invg𝐻)‘(𝐹𝑦))))
4 simp1 1059 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐻 ∈ Grp)
5 simp2 1060 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐼𝑉)
6 simp3 1061 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
7 eqid 2621 . . 3 ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
8 eqid 2621 . . 3 ( ~FG𝐼) = ( ~FG𝐼)
9 frgpup3.g . . 3 𝐺 = (freeGrp‘𝐼)
10 eqid 2621 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2621 . . 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 18109 . 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 481 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝐻 ∈ Grp)
145adantr 481 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝐼𝑉)
156adantr 481 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝐹:𝐼𝐵)
16 frgpup3.u . . . . 5 𝑈 = (varFGrp𝐼)
17 simpr 477 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ 𝑘𝐼) → 𝑘𝐼)
181, 2, 3, 13, 14, 15, 7, 8, 9, 10, 11, 16, 17frgpup2 18110 . . . 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 4704 . . 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 17585 . . . . 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 18102 . . . . 5 (𝐼𝑉𝑈:𝐼⟶(Base‘𝐺))
235, 22syl 17 . . . 4 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝑈:𝐼⟶(Base‘𝐺))
24 fcompt 6354 . . . 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 692 . . 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 6206 . . 3 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → 𝐹 = (𝑘𝐼 ↦ (𝐹𝑘)))
2719, 25, 263eqtr4d 2665 . 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 481 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝐻 ∈ Grp)
295adantr 481 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝐼𝑉)
306adantr 481 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝐹:𝐼𝐵)
31 simprl 793 . . . . 5 (((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚𝑈) = 𝐹)) → 𝑚 ∈ (𝐺 GrpHom 𝐻))
32 simprr 795 . . . . 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 18111 . . . 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 642 . . 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 2960 . 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 5239 . . . 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 2623 . . 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 3380 . 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 1323 1 ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚𝑈) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ∃!wreu 2909  c0 3891  ifcif 4058  cop 4154  cmpt 4673   I cid 4984   × cxp 5072  ran crn 5075  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  2𝑜c2o 7499  [cec 7685  Word cword 13230  Basecbs 15781   Σg cgsu 16022  Grpcgrp 17343  invgcminusg 17344   GrpHom cghm 17578   ~FG cefg 18040  freeGrpcfrgp 18041  varFGrpcvrgp 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-reverse 13244  df-s2 13530  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-0g 16023  df-gsum 16024  df-imas 16089  df-qus 16090  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-frmd 17307  df-vrmd 17308  df-grp 17346  df-minusg 17347  df-ghm 17579  df-efg 18043  df-frgp 18044  df-vrgp 18045
This theorem is referenced by:  0frgp  18113  frgpcyg  19841
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