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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 28790 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 2798 | . . . . . . 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 2798 | . . . . . . 7 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
7 | lmodvsass.t | . . . . . . 7 ⊢ × = (.r‘𝐹) | |
8 | eqid 2798 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 19632 | . . . . . 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 771 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
11 | 10 | 3expa 1115 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
12 | 11 | anabsan2 673 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
13 | 12 | exp42 439 | . 2 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))))) |
14 | 13 | 3imp2 1346 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 1rcur 19244 LModclmod 19627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-lmod 19629 |
This theorem is referenced by: lmodvs0 19661 lmodvsneg 19671 lmodsubvs 19683 lmodsubdi 19684 lmodsubdir 19685 islss3 19724 lss1d 19728 prdslmodd 19734 lmodvsinv 19801 lmhmvsca 19810 lvecvs0or 19873 lssvs0or 19875 lvecinv 19878 lspsnvs 19879 lspfixed 19893 lspsolvlem 19907 lspsolv 19908 frlmup1 20487 assa2ass 20552 ascldimul 20573 ascldimulOLD 20574 assamulgscmlem2 20586 mplmon2mul 20740 smatvscl 21129 matinv 21282 clmvsass 23694 cvsi 23735 imaslmod 30973 lshpkrlem4 36409 lcdvsass 38903 baerlem3lem1 39003 hgmapmul 39191 prjspertr 39599 mendlmod 40137 lincscm 44839 ldepsprlem 44881 lincresunit3lem3 44883 lincresunit3lem1 44888 |
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