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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 3198 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
6 | 3, 5 | jca 510 | . 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 19130 | . 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 1342 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 Grpcgrp 18855 GrpHom cghm 19127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-ghm 19128 |
This theorem is referenced by: ghmmhmb 19141 resghm 19146 conjghm 19163 qusghm 19169 invoppggim 19268 galactghm 19313 pj1ghm 19612 frgpup1 19684 mulgghm 19737 ghmfghm 19739 invghm 19742 ghmplusg 19755 ringlghm 20200 ringrghm 20201 isrnghmd 20342 isrhmd 20379 lmodvsghm 20677 pwssplit2 20815 rngqiprngghm 21058 cygznlem3 21344 psgnghm 21352 frlmup1 21572 asclghm 21656 evlslem1 21864 mat1ghm 22205 scmatghm 22255 mat2pmatghm 22452 pm2mpghm 22538 reefgim 26198 lmodvslmhm 32472 imasghm 32740 ghmquskerlem3 32805 qqhghm 33266 frlmsnic 41412 mplmapghm 41430 imasgim 42144 amgmlemALT 47937 |
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