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Mirrors > Home > MPE Home > Th. List > frgpuptf | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
frgpuptf | ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpup.a | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
2 | 1 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ 𝐵) |
3 | 2 | adantrr 715 | . . . 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 18151 | . . . . 5 ⊢ ((𝐻 ∈ Grp ∧ (𝐹‘𝑦) ∈ 𝐵) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵) |
8 | 4, 3, 7 | syl2an2r 683 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → (𝑁‘(𝐹‘𝑦)) ∈ 𝐵) |
9 | 3, 8 | ifcld 4512 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵) |
10 | 9 | ralrimivva 3191 | . 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 7766 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∅c0 4291 ifcif 4467 × cxp 5553 ⟶wf 6351 ‘cfv 6355 ∈ cmpo 7158 2oc2o 8096 Basecbs 16483 Grpcgrp 18103 invgcminusg 18104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 |
This theorem is referenced by: frgpuplem 18898 frgpupf 18899 frgpup1 18901 frgpup2 18902 frgpup3lem 18903 |
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