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Theorem frgpuptf 18906
 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 6835 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ 𝐵)
32adantrr 716 . . . 4 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → (𝐹𝑦) ∈ 𝐵)
4 frgpup.h . . . . 5 (𝜑𝐻 ∈ Grp)
5 frgpup.b . . . . . 6 𝐵 = (Base‘𝐻)
6 frgpup.n . . . . . 6 𝑁 = (invg𝐻)
75, 6grpinvcl 18161 . . . . 5 ((𝐻 ∈ Grp ∧ (𝐹𝑦) ∈ 𝐵) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
84, 3, 7syl2an2r 684 . . . 4 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
93, 8ifcld 4472 . . 3 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
109ralrimivva 3156 . 2 (𝜑 → ∀𝑦𝐼𝑧 ∈ 2o if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
11 frgpup.t . . 3 𝑇 = (𝑦𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
1211fmpo 7758 . 2 (∀𝑦𝐼𝑧 ∈ 2o if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵𝑇:(𝐼 × 2o)⟶𝐵)
1310, 12sylib 221 1 (𝜑𝑇:(𝐼 × 2o)⟶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∅c0 4245  ifcif 4427   × cxp 5520  ⟶wf 6325  ‘cfv 6329   ∈ cmpo 7144  2oc2o 8094  Basecbs 16492  Grpcgrp 18112  invgcminusg 18113 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-fv 6337  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-1st 7681  df-2nd 7682  df-0g 16724  df-mgm 17861  df-sgrp 17910  df-mnd 17921  df-grp 18115  df-minusg 18116 This theorem is referenced by:  frgpuplem  18908  frgpupf  18909  frgpup1  18911  frgpup2  18912  frgpup3lem  18913
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