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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 3199 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
6 | 3, 5 | jca 511 | . 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 19245 | . 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 1343 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 Grpcgrp 18963 GrpHom cghm 19242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-map 8866 df-ghm 19243 |
This theorem is referenced by: ghmmhmb 19257 resghm 19262 conjghm 19279 qusghm 19285 ghmqusnsg 19312 ghmquskerlem3 19316 invoppggim 19393 galactghm 19436 pj1ghm 19735 frgpup1 19807 mulgghm 19860 ghmfghm 19862 invghm 19865 ghmplusg 19878 ringlghm 20325 ringrghm 20326 isrnghmd 20467 isrhmd 20504 lmodvsghm 20937 pwssplit2 21076 rngqiprngghm 21326 cygznlem3 21605 psgnghm 21615 frlmup1 21835 asclghm 21920 evlslem1 22123 mat1ghm 22504 scmatghm 22554 mat2pmatghm 22751 pm2mpghm 22837 reefgim 26508 lmodvslmhm 33035 imasghm 33362 qqhghm 33950 aks6d1c6isolem2 42156 frlmsnic 42526 mplmapghm 42542 imasgim 43088 amgmlemALT 49033 |
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