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Theorem frgpuptf 18536
 Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpup.b 𝐵 = (Base‘𝐻)
frgpup.n 𝑁 = (invg𝐻)
frgpup.t 𝑇 = (𝑦𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
frgpup.h (𝜑𝐻 ∈ Grp)
frgpup.i (𝜑𝐼𝑉)
frgpup.a (𝜑𝐹:𝐼𝐵)
Assertion
Ref Expression
frgpuptf (𝜑𝑇:(𝐼 × 2o)⟶𝐵)
Distinct variable groups:   𝑦,𝑧,𝐹   𝑦,𝑁,𝑧   𝑦,𝐵,𝑧   𝜑,𝑦,𝑧   𝑦,𝐼,𝑧
Allowed substitution hints:   𝑇(𝑦,𝑧)   𝐻(𝑦,𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem frgpuptf
StepHypRef Expression
1 frgpup.a . . . . . 6 (𝜑𝐹:𝐼𝐵)
21ffvelrnda 6608 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ 𝐵)
32adantrr 710 . . . 4 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → (𝐹𝑦) ∈ 𝐵)
4 frgpup.h . . . . . 6 (𝜑𝐻 ∈ Grp)
54adantr 474 . . . . 5 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → 𝐻 ∈ Grp)
6 frgpup.b . . . . . 6 𝐵 = (Base‘𝐻)
7 frgpup.n . . . . . 6 𝑁 = (invg𝐻)
86, 7grpinvcl 17821 . . . . 5 ((𝐻 ∈ Grp ∧ (𝐹𝑦) ∈ 𝐵) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
95, 3, 8syl2anc 581 . . . 4 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
103, 9ifcld 4351 . . 3 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
1110ralrimivva 3180 . 2 (𝜑 → ∀𝑦𝐼𝑧 ∈ 2o if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
12 frgpup.t . . 3 𝑇 = (𝑦𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
1312fmpt2 7500 . 2 (∀𝑦𝐼𝑧 ∈ 2o if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵𝑇:(𝐼 × 2o)⟶𝐵)
1411, 13sylib 210 1 (𝜑𝑇:(𝐼 × 2o)⟶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ∅c0 4144  ifcif 4306   × cxp 5340  ⟶wf 6119  ‘cfv 6123   ↦ cmpt2 6907  2oc2o 7820  Basecbs 16222  Grpcgrp 17776  invgcminusg 17777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780 This theorem is referenced by:  frgpuplem  18538  frgpupf  18539  frgpup1  18541  frgpup2  18542  frgpup3lem  18543
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