Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lmodvsass | Structured version Visualization version GIF version |
Description: Associative law for scalar product. (ax-hvmulass 28786 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 2823 | . . . . . . 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 2823 | . . . . . . 7 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
7 | lmodvsass.t | . . . . . . 7 ⊢ × = (.r‘𝐹) | |
8 | eqid 2823 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 19641 | . . . . . 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 770 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
11 | 10 | 3expa 1114 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
12 | 11 | anabsan2 672 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
13 | 12 | exp42 438 | . 2 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))))) |
14 | 13 | 3imp2 1345 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 1rcur 19253 LModclmod 19636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-lmod 19638 |
This theorem is referenced by: lmodvs0 19670 lmodvsneg 19680 lmodsubvs 19692 lmodsubdi 19693 lmodsubdir 19694 islss3 19733 lss1d 19737 prdslmodd 19743 lmodvsinv 19810 lmhmvsca 19819 lvecvs0or 19882 lssvs0or 19884 lvecinv 19887 lspsnvs 19888 lspfixed 19902 lspsolvlem 19916 lspsolv 19917 assa2ass 20097 ascldimul 20118 ascldimulOLD 20119 assamulgscmlem2 20131 mplmon2mul 20283 frlmup1 20944 smatvscl 21135 matinv 21288 clmvsass 23695 cvsi 23736 imaslmod 30924 lshpkrlem4 36251 lcdvsass 38745 baerlem3lem1 38845 hgmapmul 39033 prjspertr 39262 mendlmod 39800 lincscm 44492 ldepsprlem 44534 lincresunit3lem3 44536 lincresunit3lem1 44541 |
Copyright terms: Public domain | W3C validator |