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| Mirrors > Home > MPE Home > Th. List > lmodvsdir | Structured version Visualization version GIF version | ||
| Description: Distributive law for scalar product (right-distributivity). (ax-hvdistr1 31028 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| lmodvsdir.v | ⊢ 𝑉 = (Base‘𝑊) | 
| lmodvsdir.a | ⊢ + = (+g‘𝑊) | 
| lmodvsdir.f | ⊢ 𝐹 = (Scalar‘𝑊) | 
| lmodvsdir.s | ⊢ · = ( ·𝑠 ‘𝑊) | 
| lmodvsdir.k | ⊢ 𝐾 = (Base‘𝐹) | 
| lmodvsdir.p | ⊢ ⨣ = (+g‘𝐹) | 
| Ref | Expression | 
|---|---|
| lmodvsdir | ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lmodvsdir.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
| 2 | lmodvsdir.a | . . . . . . . 8 ⊢ + = (+g‘𝑊) | |
| 3 | lmodvsdir.s | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 4 | lmodvsdir.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 5 | lmodvsdir.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
| 6 | lmodvsdir.p | . . . . . . . 8 ⊢ ⨣ = (+g‘𝐹) | |
| 7 | eqid 2736 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
| 8 | eqid 2736 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 20864 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) ∧ (((𝑄(.r‘𝐹)𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋))) | 
| 10 | 9 | simpld 494 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))) | 
| 11 | 10 | simp3d 1144 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | 
| 12 | 11 | 3expa 1118 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | 
| 13 | 12 | anabsan2 674 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | 
| 14 | 13 | exp42 435 | . 2 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))))) | 
| 15 | 14 | 3imp2 1349 | 1 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 1rcur 20179 LModclmod 20859 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-lmod 20861 | 
| This theorem is referenced by: lmod0vs 20894 lmodvsmmulgdi 20896 lmodvneg1 20904 lmodcom 20907 lmodsubdir 20919 islss3 20958 lss1d 20962 prdslmodd 20968 lspsolvlem 21145 frlmup1 21819 asclghm 21904 scmataddcl 22523 scmatghm 22540 pm2mpghm 22823 clmvsdir 25125 cvsi 25164 lmodvslmhm 33054 imaslmod 33382 lshpkrlem4 39115 baerlem3lem1 41710 baerlem5blem1 41712 hgmapadd 41897 mendlmod 43206 lmodvsmdi 48300 lincsum 48351 ldepsprlem 48394 | 
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