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Theorem frgpuptf 19548
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)
21ffvelcdmda 7032 . . . . 5 ((𝜑𝑦𝐼) → (𝐹𝑦) ∈ 𝐵)
32adantrr 715 . . . 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 18795 . . . . 5 ((𝐻 ∈ Grp ∧ (𝐹𝑦) ∈ 𝐵) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
84, 3, 7syl2an2r 683 . . . 4 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → (𝑁‘(𝐹𝑦)) ∈ 𝐵)
93, 8ifcld 4531 . . 3 ((𝜑 ∧ (𝑦𝐼𝑧 ∈ 2o)) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
109ralrimivva 3196 . 2 (𝜑 → ∀𝑦𝐼𝑧 ∈ 2o if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵)
11 frgpup.t . . 3 𝑇 = (𝑦𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
1211fmpo 7997 . 2 (∀𝑦𝐼𝑧 ∈ 2o if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) ∈ 𝐵𝑇:(𝐼 × 2o)⟶𝐵)
1310, 12sylib 217 1 (𝜑𝑇:(𝐼 × 2o)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3063  c0 4281  ifcif 4485   × cxp 5630  wf 6490  cfv 6494  cmpo 7356  2oc2o 8403  Basecbs 17080  Grpcgrp 18745  invgcminusg 18746
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-riota 7310  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7918  df-2nd 7919  df-0g 17320  df-mgm 18494  df-sgrp 18543  df-mnd 18554  df-grp 18748  df-minusg 18749
This theorem is referenced by:  frgpuplem  19550  frgpupf  19551  frgpup1  19553  frgpup2  19554  frgpup3lem  19555
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