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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 2798 | . . . . . 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 19792 | . . . . 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 500 | . . . 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 1141 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏))) |
10 | fvoveq1 7158 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏))) | |
11 | oveq1 7142 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑏))) | |
12 | 10, 11 | eqeq12d 2814 | . . . 4 ⊢ (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)))) |
13 | oveq2 7143 | . . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌)) | |
14 | 13 | fveq2d 6649 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌))) |
15 | fveq2 6645 | . . . . . 6 ⊢ (𝑏 = 𝑌 → (𝐹‘𝑏) = (𝐹‘𝑌)) | |
16 | 15 | oveq2d 7151 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑌))) |
17 | 14, 16 | eqeq12d 2814 | . . . 4 ⊢ (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))) |
18 | 12, 17 | rspc2v 3581 | . . 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 1113 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 GrpHom cghm 18347 LModclmod 19627 LMHom clmhm 19784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-lmhm 19787 |
This theorem is referenced by: islmhm2 19803 lmhmco 19808 lmhmplusg 19809 lmhmvsca 19810 lmhmf1o 19811 lmhmima 19812 lmhmpreima 19813 reslmhm 19817 reslmhm2 19818 reslmhm2b 19819 lmhmeql 19820 ipass 20334 lindfmm 20516 nmoleub2lem3 23720 nmoleub3 23724 mendassa 40138 |
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