![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | ffvelcdmda 7032 | . . . . 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 18795 | . . . . 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 4531 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵) |
10 | 9 | ralrimivva 3196 | . 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 7997 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) ∈ 𝐵 ↔ 𝑇:(𝐼 × 2o)⟶𝐵) |
13 | 10, 12 | sylib 217 | 1 ⊢ (𝜑 → 𝑇:(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 |
Copyright terms: Public domain | W3C validator |