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| Mirrors > Home > MPE Home > Th. List > lmhmlin | Structured version Visualization version GIF version | ||
| Description: A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
| Ref | Expression |
|---|---|
| lmhmlin.k | ⊢ 𝐾 = (Scalar‘𝑆) |
| lmhmlin.b | ⊢ 𝐵 = (Base‘𝐾) |
| lmhmlin.e | ⊢ 𝐸 = (Base‘𝑆) |
| lmhmlin.m | ⊢ · = ( ·𝑠 ‘𝑆) |
| lmhmlin.n | ⊢ × = ( ·𝑠 ‘𝑇) |
| Ref | Expression |
|---|---|
| lmhmlin | ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lmhmlin.k | . . . . . 6 ⊢ 𝐾 = (Scalar‘𝑆) | |
| 2 | eqid 2737 | . . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
| 3 | lmhmlin.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 4 | lmhmlin.e | . . . . . 6 ⊢ 𝐸 = (Base‘𝑆) | |
| 5 | lmhmlin.m | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑆) | |
| 6 | lmhmlin.n | . . . . . 6 ⊢ × = ( ·𝑠 ‘𝑇) | |
| 7 | 1, 2, 3, 4, 5, 6 | islmhm 21026 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏))))) |
| 8 | 7 | simprbi 496 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)))) |
| 9 | 8 | simp3d 1145 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏))) |
| 10 | fvoveq1 7454 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏))) | |
| 11 | oveq1 7438 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑏))) | |
| 12 | 10, 11 | eqeq12d 2753 | . . . 4 ⊢ (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)))) |
| 13 | oveq2 7439 | . . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌)) | |
| 14 | 13 | fveq2d 6910 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌))) |
| 15 | fveq2 6906 | . . . . . 6 ⊢ (𝑏 = 𝑌 → (𝐹‘𝑏) = (𝐹‘𝑌)) | |
| 16 | 15 | oveq2d 7447 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑌))) |
| 17 | 14, 16 | eqeq12d 2753 | . . . 4 ⊢ (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))) |
| 18 | 12, 17 | rspc2v 3633 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))) |
| 19 | 9, 18 | syl5com 31 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))) |
| 20 | 19 | 3impib 1117 | 1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 GrpHom cghm 19230 LModclmod 20858 LMHom clmhm 21018 |
| 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-pr 5432 |
| 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-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-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-lmhm 21021 |
| This theorem is referenced by: islmhm2 21037 lmhmco 21042 lmhmplusg 21043 lmhmvsca 21044 lmhmf1o 21045 lmhmima 21046 lmhmpreima 21047 reslmhm 21051 reslmhm2 21052 reslmhm2b 21053 lmhmeql 21054 ipass 21663 lindfmm 21847 nmoleub2lem3 25148 nmoleub3 25152 lmhmimasvsca 33044 lmhmqusker 33445 mendassa 43202 |
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