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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 2735 | . . . . . 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 21044 | . . . . 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 1143 | . . 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 2751 | . . . 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 6911 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌))) |
15 | fveq2 6907 | . . . . . 6 ⊢ (𝑏 = 𝑌 → (𝐹‘𝑏) = (𝐹‘𝑌)) | |
16 | 15 | oveq2d 7447 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑌))) |
17 | 14, 16 | eqeq12d 2751 | . . . 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 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 GrpHom cghm 19243 LModclmod 20875 LMHom clmhm 21036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-lmhm 21039 |
This theorem is referenced by: islmhm2 21055 lmhmco 21060 lmhmplusg 21061 lmhmvsca 21062 lmhmf1o 21063 lmhmima 21064 lmhmpreima 21065 reslmhm 21069 reslmhm2 21070 reslmhm2b 21071 lmhmeql 21072 ipass 21681 lindfmm 21865 nmoleub2lem3 25162 nmoleub3 25166 lmhmimasvsca 33027 lmhmqusker 33425 mendassa 43179 |
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