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 = (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 6524	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ 𝐵)
3	2	adantrr 755	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → (𝐹‘𝑦) ∈ 𝐵)
4		frgpup.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Grp)
5	4	adantr 472	. . . . 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 17689	. . . . 5 ⊢ ((𝐻 ∈ Grp ∧ (𝐹‘𝑦) ∈ 𝐵) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵)
9	5, 3, 8	syl2anc 696	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵)
10	3, 9	ifcld 4276	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵)
11	10	ralrimivva 3110	. 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 7407	. 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵 ↔ 𝑇:(𝐼 × 2𝑜)⟶𝐵)
14	11, 13	sylib 208	1 ⊢ (𝜑 → 𝑇:(𝐼 × 2𝑜)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2140  ∀wral 3051  ∅c0 4059  ifcif 4231   × cxp 5265  ⟶wf 6046  ‘cfv 6050   ↦ cmpt2 6817  2_𝑜c2o 7725  Basecbs 16080  Grpcgrp 17644  inv_gcminusg 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