Theorem frgpuptf 19810

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	ffvelcdmda 7065	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ 𝐵)
3	2	adantrr 727	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → (𝐹‘𝑦) ∈ 𝐵)
4		frgpup.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ Grp)
5		frgpup.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐻)
6		frgpup.n	. . . . . 6 ⊢ 𝑁 = (invg‘𝐻)
7	5, 6	grpinvcl 19029	. . . . 5 ⊢ ((𝐻 ∈ Grp ∧ (𝐹‘𝑦) ∈ 𝐵) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵)
8	4, 3, 7	syl2an2r 695	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵)
9	3, 8	ifcld 4527	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵)
10	9	ralrimivva 3205	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵)
11		frgpup.t	. . 3 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
12	11	fmpo 8049	. 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵 ↔ 𝑇:(𝐼 × 2o)⟶𝐵)
13	10, 12	sylib 220	1 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∅c0 4285  ifcif 4480   × cxp 5645  ⟶wf 6517  ‘cfv 6521   ∈ cmpo 7398  2_oc2o 8431  Basecbs 17245  Grpcgrp 18975  inv_gcminusg 18976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-0g 17470  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-grp 18978  df-minusg 18979

This theorem is referenced by:  frgpuplem  19812  frgpupf  19813  frgpup1  19815  frgpup2  19816  frgpup3lem  19817