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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 28712 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 2818 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
8 | eqid 2818 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 19568 | . . . . . . 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 495 | . . . . . 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 1136 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
12 | 11 | 3expa 1110 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
13 | 12 | anabsan2 670 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
14 | 13 | exp42 436 | . 2 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))))) |
15 | 14 | 3imp2 1341 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 1rcur 19180 LModclmod 19563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-lmod 19565 |
This theorem is referenced by: lmod0vs 19596 lmodvsmmulgdi 19598 lmodvneg1 19606 lmodcom 19609 lmodsubdir 19621 islss3 19660 lss1d 19664 prdslmodd 19670 lspsolvlem 19843 asclghm 20040 frlmup1 20870 scmataddcl 21053 scmatghm 21070 pm2mpghm 21352 clmvsdir 23622 cvsi 23661 lmodvslmhm 30615 imaslmod 30849 lshpkrlem4 36129 baerlem3lem1 38723 baerlem5blem1 38725 hgmapadd 38910 mendlmod 39671 lmodvsmdi 44358 lincsum 44412 ldepsprlem 44455 |
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