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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 3202 | . . 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 19233 | . 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 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 GrpHom cghm 19230 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-ghm 19231 |
| This theorem is referenced by: ghmmhmb 19245 resghm 19250 conjghm 19267 qusghm 19273 ghmqusnsg 19300 ghmquskerlem3 19304 invoppggim 19379 galactghm 19422 pj1ghm 19721 frgpup1 19793 mulgghm 19846 ghmfghm 19848 invghm 19851 ghmplusg 19864 ringlghm 20309 ringrghm 20310 isrnghmd 20451 isrhmd 20488 lmodvsghm 20921 pwssplit2 21059 rngqiprngghm 21309 cygznlem3 21588 psgnghm 21598 frlmup1 21818 asclghm 21903 evlslem1 22106 mat1ghm 22489 scmatghm 22539 mat2pmatghm 22736 pm2mpghm 22822 reefgim 26494 lmodvslmhm 33053 imasghm 33383 qqhghm 33989 aks6d1c6isolem2 42176 frlmsnic 42550 mplmapghm 42566 imasgim 43112 amgmlemALT 49322 |
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