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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 3201 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
6 | 3, 5 | jca 513 | . 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 19086 | . 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 1345 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 +gcplusg 17193 Grpcgrp 18815 GrpHom cghm 19083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-ghm 19084 |
This theorem is referenced by: ghmmhmb 19097 resghm 19102 conjghm 19117 qusghm 19123 invoppggim 19220 galactghm 19265 pj1ghm 19564 frgpup1 19636 mulgghm 19688 ghmfghm 19690 invghm 19693 ghmplusg 19706 ringlghm 20114 ringrghm 20115 isrhmd 20255 lmodvsghm 20521 pwssplit2 20659 cygznlem3 21109 psgnghm 21117 frlmup1 21337 asclghm 21419 evlslem1 21627 mat1ghm 21967 scmatghm 22017 mat2pmatghm 22214 pm2mpghm 22300 reefgim 25944 lmodvslmhm 32180 ghmqusker 32494 qqhghm 32906 frlmsnic 41060 mplmapghm 41078 imasgim 41775 isrnghmd 46634 rngqiprngghm 46713 amgmlemALT 47752 |
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