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Mirrors > Home > MPE Home > Th. List > islmhmd | Structured version Visualization version GIF version |
Description: Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
Ref | Expression |
---|---|
islmhmd.x | ⊢ 𝑋 = (Base‘𝑆) |
islmhmd.a | ⊢ · = ( ·𝑠 ‘𝑆) |
islmhmd.b | ⊢ × = ( ·𝑠 ‘𝑇) |
islmhmd.k | ⊢ 𝐾 = (Scalar‘𝑆) |
islmhmd.j | ⊢ 𝐽 = (Scalar‘𝑇) |
islmhmd.n | ⊢ 𝑁 = (Base‘𝐾) |
islmhmd.s | ⊢ (𝜑 → 𝑆 ∈ LMod) |
islmhmd.t | ⊢ (𝜑 → 𝑇 ∈ LMod) |
islmhmd.c | ⊢ (𝜑 → 𝐽 = 𝐾) |
islmhmd.f | ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
islmhmd.l | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) |
Ref | Expression |
---|---|
islmhmd | ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islmhmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ LMod) | |
2 | islmhmd.t | . 2 ⊢ (𝜑 → 𝑇 ∈ LMod) | |
3 | islmhmd.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
4 | islmhmd.c | . . 3 ⊢ (𝜑 → 𝐽 = 𝐾) | |
5 | islmhmd.l | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) | |
6 | 5 | ralrimivva 3201 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) |
7 | 3, 4, 6 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))) |
8 | islmhmd.k | . . 3 ⊢ 𝐾 = (Scalar‘𝑆) | |
9 | islmhmd.j | . . 3 ⊢ 𝐽 = (Scalar‘𝑇) | |
10 | islmhmd.n | . . 3 ⊢ 𝑁 = (Base‘𝐾) | |
11 | islmhmd.x | . . 3 ⊢ 𝑋 = (Base‘𝑆) | |
12 | islmhmd.a | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
13 | islmhmd.b | . . 3 ⊢ × = ( ·𝑠 ‘𝑇) | |
14 | 8, 9, 10, 11, 12, 13 | islmhm 20615 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
15 | 1, 2, 7, 14 | syl21anbrc 1345 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 Scalarcsca 17187 ·𝑠 cvsca 17188 GrpHom cghm 19074 LModclmod 20448 LMHom clmhm 20607 |
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-sep 5295 ax-nul 5302 ax-pr 5423 |
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-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6487 df-fun 6537 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-lmhm 20610 |
This theorem is referenced by: 0lmhm 20628 idlmhm 20629 invlmhm 20630 lmhmco 20631 lmhmplusg 20632 lmhmvsca 20633 lmhmf1o 20634 reslmhm2 20641 reslmhm2b 20642 pwsdiaglmhm 20645 pwssplit3 20649 frlmup1 21326 quslmhm 32432 lmhmqusker 32489 frlmsnic 41014 |
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