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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 2738 | . . . . . 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 20289 | . . . . 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 497 | . . . 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 1143 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏))) |
10 | fvoveq1 7298 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏))) | |
11 | oveq1 7282 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑏))) | |
12 | 10, 11 | eqeq12d 2754 | . . . 4 ⊢ (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)))) |
13 | oveq2 7283 | . . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌)) | |
14 | 13 | fveq2d 6778 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌))) |
15 | fveq2 6774 | . . . . . 6 ⊢ (𝑏 = 𝑌 → (𝐹‘𝑏) = (𝐹‘𝑌)) | |
16 | 15 | oveq2d 7291 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑌))) |
17 | 14, 16 | eqeq12d 2754 | . . . 4 ⊢ (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))) |
18 | 12, 17 | rspc2v 3570 | . . 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 1115 | 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Scalarcsca 16965 ·𝑠 cvsca 16966 GrpHom cghm 18831 LModclmod 20123 LMHom clmhm 20281 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-lmhm 20284 |
This theorem is referenced by: islmhm2 20300 lmhmco 20305 lmhmplusg 20306 lmhmvsca 20307 lmhmf1o 20308 lmhmima 20309 lmhmpreima 20310 reslmhm 20314 reslmhm2 20315 reslmhm2b 20316 lmhmeql 20317 ipass 20850 lindfmm 21034 nmoleub2lem3 24278 nmoleub3 24282 mendassa 41019 |
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