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| Mirrors > Home > MPE Home > Th. List > lmodvsass | Structured version Visualization version GIF version | ||
| Description: Associative law for scalar product. (ax-hvmulass 31026 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
| Ref | Expression |
|---|---|
| lmodvsass.v | ⊢ 𝑉 = (Base‘𝑊) |
| lmodvsass.f | ⊢ 𝐹 = (Scalar‘𝑊) |
| lmodvsass.s | ⊢ · = ( ·𝑠 ‘𝑊) |
| lmodvsass.k | ⊢ 𝐾 = (Base‘𝐹) |
| lmodvsass.t | ⊢ × = (.r‘𝐹) |
| Ref | Expression |
|---|---|
| lmodvsass | ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lmodvsass.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
| 2 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 3 | lmodvsass.s | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 4 | lmodvsass.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 5 | lmodvsass.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
| 6 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
| 7 | lmodvsass.t | . . . . . . 7 ⊢ × = (.r‘𝐹) | |
| 8 | eqid 2737 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 20863 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g‘𝑊)𝑋)) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋)) ∧ ((𝑄(+g‘𝐹)𝑅) · 𝑋) = ((𝑄 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋))) ∧ (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋))) |
| 10 | 9 | simprld 772 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
| 11 | 10 | 3expa 1119 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
| 12 | 11 | anabsan2 674 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
| 13 | 12 | exp42 435 | . 2 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))))) |
| 14 | 13 | 3imp2 1350 | 1 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 1rcur 20178 LModclmod 20858 |
| 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-ext 2708 ax-nul 5306 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lmod 20860 |
| This theorem is referenced by: lmodvs0 20894 lmodvsneg 20904 lmodsubvs 20916 lmodsubdi 20917 lmodsubdir 20918 islss3 20957 lss1d 20961 prdslmodd 20967 lmodvsinv 21035 lmhmvsca 21044 lvecvs0or 21110 lssvs0or 21112 lvecinv 21115 lspsnvs 21116 lspfixed 21130 lspsolvlem 21144 lspsolv 21145 frlmup1 21818 assa2ass 21883 assa2ass2 21884 ascldimul 21908 assamulgscmlem2 21920 mplmon2mul 22093 smatvscl 22530 matinv 22683 clmvsass 25122 cvsi 25163 imaslmod 33381 lshpkrlem4 39114 lcdvsass 41609 baerlem3lem1 41709 hgmapmul 41897 prjspertr 42615 prjspner1 42636 mendlmod 43201 lincscm 48347 ldepsprlem 48389 lincresunit3lem3 48391 lincresunit3lem1 48396 |
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