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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 3191 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) |
7 | 3, 4, 6 | 3jca 1124 | . 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 19799 | . 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 1340 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 GrpHom cghm 18355 LModclmod 19634 LMHom clmhm 19791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-lmhm 19794 |
This theorem is referenced by: 0lmhm 19812 idlmhm 19813 invlmhm 19814 lmhmco 19815 lmhmplusg 19816 lmhmvsca 19817 lmhmf1o 19818 reslmhm2 19825 reslmhm2b 19826 pwsdiaglmhm 19829 pwssplit3 19833 frlmup1 20942 quslmhm 30924 frlmsnic 39169 |
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