Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isghmd | Structured version Visualization version GIF version |
Description: Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
Ref | Expression |
---|---|
isghmd.x | ⊢ 𝑋 = (Base‘𝑆) |
isghmd.y | ⊢ 𝑌 = (Base‘𝑇) |
isghmd.a | ⊢ + = (+g‘𝑆) |
isghmd.b | ⊢ ⨣ = (+g‘𝑇) |
isghmd.s | ⊢ (𝜑 → 𝑆 ∈ Grp) |
isghmd.t | ⊢ (𝜑 → 𝑇 ∈ Grp) |
isghmd.f | ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
isghmd.l | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
Ref | Expression |
---|---|
isghmd | ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isghmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ Grp) | |
2 | isghmd.t | . 2 ⊢ (𝜑 → 𝑇 ∈ Grp) | |
3 | isghmd.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) | |
4 | isghmd.l | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
5 | 4 | ralrimivva 3191 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
6 | 3, 5 | jca 514 | . 2 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))) |
7 | isghmd.x | . . 3 ⊢ 𝑋 = (Base‘𝑆) | |
8 | isghmd.y | . . 3 ⊢ 𝑌 = (Base‘𝑇) | |
9 | isghmd.a | . . 3 ⊢ + = (+g‘𝑆) | |
10 | isghmd.b | . . 3 ⊢ ⨣ = (+g‘𝑇) | |
11 | 7, 8, 9, 10 | isghm 18352 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))) |
12 | 1, 2, 6, 11 | syl21anbrc 1340 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Grpcgrp 18097 GrpHom cghm 18349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-ghm 18350 |
This theorem is referenced by: ghmmhmb 18363 resghm 18368 conjghm 18383 qusghm 18389 invoppggim 18482 galactghm 18526 pj1ghm 18823 frgpup1 18895 mulgghm 18943 ghmfghm 18945 invghm 18948 ghmplusg 18960 ringlghm 19348 ringrghm 19349 isrhmd 19475 lmodvsghm 19689 pwssplit2 19826 asclghm 20106 evlslem1 20289 cygznlem3 20710 psgnghm 20718 frlmup1 20936 mat1ghm 21086 scmatghm 21136 mat2pmatghm 21332 pm2mpghm 21418 reefgim 25032 lmodvslmhm 30683 qqhghm 31224 frlmsnic 39142 imasgim 39693 isrnghmd 44166 amgmlemALT 44897 |
Copyright terms: Public domain | W3C validator |