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Theorem frgpuptf 18404
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 6524 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ 𝐵)
32adantrr 755 . . . 4 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → (𝐹𝑦) ∈ 𝐵)
4 frgpup.h . . . . . 6 (𝜑𝐻 ∈ Grp)
54adantr 472 . . . . 5 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝐻 ∈ Grp)
6 frgpup.b . . . . . 6 𝐵 = (Base‘𝐻)
7 frgpup.n . . . . . 6 𝑁 = (invg𝐻)
86, 7grpinvcl 17689 . . . . 5 ((𝐻 ∈ Grp ∧ (𝐹𝑦) ∈ 𝐵) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
95, 3, 8syl2anc 696 . . . 4 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
103, 9ifcld 4276 . . 3 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
1110ralrimivva 3110 . 2 (𝜑 → ∀𝑦𝐼𝑧 ∈ 2𝑜 if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
12 frgpup.t . . 3 𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
1312fmpt2 7407 . 2 (∀𝑦𝐼𝑧 ∈ 2𝑜 if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵𝑇:(𝐼 × 2𝑜)⟶𝐵)
1411, 13sylib 208 1 (𝜑𝑇:(𝐼 × 2𝑜)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  wral 3051  c0 4059  ifcif 4231   × cxp 5265  wf 6046  cfv 6050  cmpt2 6817  2𝑜c2o 7725  Basecbs 16080  Grpcgrp 17644  invgcminusg 17645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-0g 16325  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-grp 17647  df-minusg 17648
This theorem is referenced by:  frgpuplem  18406  frgpupf  18407  frgpup1  18409  frgpup2  18410  frgpup3lem  18411
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