Theorem frgpuptf 18006

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 = (invg‘𝐻)
frgpup.t	⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
frgpup.h	⊢ (𝜑 → 𝐻 ∈ Grp)
frgpup.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
frgpup.a	⊢ (𝜑 → 𝐹:𝐼⟶𝐵)

Assertion

Ref	Expression
frgpuptf	⊢ (𝜑 → 𝑇:(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

Step	Hyp	Ref	Expression
1		frgpup.a	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
2	1	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ 𝐵)
3	2	adantrr 749	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → (𝐹‘𝑦) ∈ 𝐵)
4		frgpup.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Grp)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝐻 ∈ Grp)
6		frgpup.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐻)
7		frgpup.n	. . . . . 6 ⊢ 𝑁 = (invg‘𝐻)
8	6, 7	grpinvcl 17290	. . . . 5 ⊢ ((𝐻 ∈ Grp ∧ (𝐹‘𝑦) ∈ 𝐵) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵)
9	5, 3, 8	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵)
10	3, 9	ifcld 4081	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵)
11	10	ralrimivva 2954	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵)
12		frgpup.t	. . 3 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
13	12	fmpt2 7126	. 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵 ↔ 𝑇:(𝐼 × 2𝑜)⟶𝐵)
14	11, 13	sylib 207	1 ⊢ (𝜑 → 𝑇:(𝐼 × 2𝑜)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874  ifcif 4036   × cxp 5036  ⟶wf 5800  ‘cfv 5804   ↦ cmpt2 6551  2_𝑜c2o 7441  Basecbs 15695  Grpcgrp 17245  inv_gcminusg 17246

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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-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-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-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-1st 7059  df-2nd 7060  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249

This theorem is referenced by:  frgpuplem  18008  frgpupf  18009  frgpup1  18011  frgpup2  18012  frgpup3lem  18013