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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 | . . 3 ⊢ (𝜑 → 𝑆 ∈ LMod) | |
2 | islmhmd.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ LMod) | |
3 | 1, 2 | jca 553 | . 2 ⊢ (𝜑 → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod)) |
4 | islmhmd.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
5 | islmhmd.c | . . 3 ⊢ (𝜑 → 𝐽 = 𝐾) | |
6 | islmhmd.l | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) | |
7 | 6 | ralrimivva 2954 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) |
8 | 4, 5, 7 | 3jca 1235 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))) |
9 | islmhmd.k | . . 3 ⊢ 𝐾 = (Scalar‘𝑆) | |
10 | islmhmd.j | . . 3 ⊢ 𝐽 = (Scalar‘𝑇) | |
11 | islmhmd.n | . . 3 ⊢ 𝑁 = (Base‘𝐾) | |
12 | islmhmd.x | . . 3 ⊢ 𝑋 = (Base‘𝑆) | |
13 | islmhmd.a | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
14 | islmhmd.b | . . 3 ⊢ × = ( ·𝑠 ‘𝑇) | |
15 | 9, 10, 11, 12, 13, 14 | islmhm 18848 | . 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))))) |
16 | 3, 8, 15 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 GrpHom cghm 17480 LModclmod 18686 LMHom clmhm 18840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-lmhm 18843 |
This theorem is referenced by: 0lmhm 18861 idlmhm 18862 invlmhm 18863 lmhmco 18864 lmhmplusg 18865 lmhmvsca 18866 lmhmf1o 18867 reslmhm2 18874 reslmhm2b 18875 pwsdiaglmhm 18878 pwssplit3 18882 frlmup1 19956 |
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