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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 31036 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 2735 | . . . . . . 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 2735 | . . . . . . 7 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
7 | lmodvsass.t | . . . . . . 7 ⊢ × = (.r‘𝐹) | |
8 | eqid 2735 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 20880 | . . . . . 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 1117 | . . . 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 1348 | 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 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 1rcur 20199 LModclmod 20875 |
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-ext 2706 ax-nul 5312 |
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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-lmod 20877 |
This theorem is referenced by: lmodvs0 20911 lmodvsneg 20921 lmodsubvs 20933 lmodsubdi 20934 lmodsubdir 20935 islss3 20975 lss1d 20979 prdslmodd 20985 lmodvsinv 21053 lmhmvsca 21062 lvecvs0or 21128 lssvs0or 21130 lvecinv 21133 lspsnvs 21134 lspfixed 21148 lspsolvlem 21162 lspsolv 21163 frlmup1 21836 assa2ass 21901 assa2ass2 21902 ascldimul 21926 assamulgscmlem2 21938 mplmon2mul 22111 smatvscl 22546 matinv 22699 clmvsass 25136 cvsi 25177 imaslmod 33361 lshpkrlem4 39095 lcdvsass 41590 baerlem3lem1 41690 hgmapmul 41878 prjspertr 42592 prjspner1 42613 mendlmod 43178 lincscm 48276 ldepsprlem 48318 lincresunit3lem3 48320 lincresunit3lem1 48325 |
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