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Mirrors > Home > MPE Home > Th. List > lmod0vs | Structured version Visualization version GIF version |
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31038 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmod0vs.v | ⊢ 𝑉 = (Base‘𝑊) |
lmod0vs.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmod0vs.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmod0vs.o | ⊢ 𝑂 = (0g‘𝐹) |
lmod0vs.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lmod0vs | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ LMod) | |
2 | lmod0vs.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 20882 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Ring) |
5 | eqid 2734 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | lmod0vs.o | . . . . . . 7 ⊢ 𝑂 = (0g‘𝐹) | |
7 | 5, 6 | ring0cl 20280 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝑂 ∈ (Base‘𝐹)) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹)) |
9 | simpr 484 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
10 | lmod0vs.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | eqid 2734 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
12 | lmod0vs.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
13 | eqid 2734 | . . . . . 6 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
14 | 10, 11, 2, 12, 5, 13 | lmodvsdir 20900 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑂 ∈ (Base‘𝐹) ∧ 𝑂 ∈ (Base‘𝐹) ∧ 𝑋 ∈ 𝑉)) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) |
15 | 1, 8, 8, 9, 14 | syl13anc 1371 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) |
16 | ringgrp 20255 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
17 | 4, 16 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Grp) |
18 | 5, 13, 6 | grplid 18997 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑂 ∈ (Base‘𝐹)) → (𝑂(+g‘𝐹)𝑂) = 𝑂) |
19 | 17, 8, 18 | syl2anc 584 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂(+g‘𝐹)𝑂) = 𝑂) |
20 | 19 | oveq1d 7445 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = (𝑂 · 𝑋)) |
21 | 15, 20 | eqtr3d 2776 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋)) |
22 | 10, 2, 12, 5 | lmodvscl 20892 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑂 ∈ (Base‘𝐹) ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
23 | 1, 8, 9, 22 | syl3anc 1370 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
24 | lmod0vs.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
25 | 10, 11, 24 | lmod0vid 20908 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑂 · 𝑋) ∈ 𝑉) → (((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋) ↔ 0 = (𝑂 · 𝑋))) |
26 | 23, 25 | syldan 591 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋) ↔ 0 = (𝑂 · 𝑋))) |
27 | 21, 26 | mpbid 232 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 0 = (𝑂 · 𝑋)) |
28 | 27 | eqcomd 2740 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17485 Grpcgrp 18963 Ringcrg 20250 LModclmod 20874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-riota 7387 df-ov 7433 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-ring 20252 df-lmod 20876 |
This theorem is referenced by: lmodvs0 20910 lmodvsmmulgdi 20911 lcomfsupp 20916 lmodvneg1 20919 mptscmfsupp0 20941 lvecvs0or 21127 lssvs0or 21129 lspsneleq 21134 lspdisj 21144 lspfixed 21147 lspexch 21148 lspsolvlem 21161 lspsolv 21162 uvcresum 21830 frlmsslsp 21833 frlmup1 21835 frlmup2 21836 ascl0 21921 mplcoe1 22072 mplbas2 22077 ply10s0 22274 ply1scl0OLD 22309 gsummoncoe1 22327 evls1fpws 22388 pmatcollpwscmatlem1 22810 idpm2idmp 22822 mp2pm2mplem4 22830 pm2mpmhmlem1 22839 monmat2matmon 22845 cpmidpmatlem3 22893 clm0vs 25141 plypf1 26265 lmodslmd 33192 r1p0 33605 ply1degltdimlem 33649 lbsdiflsp0 33653 fedgmullem2 33657 lshpkrlem1 39091 ldual0vs 39141 lclkrlem1 41488 lcd0vs 41597 baerlem3lem1 41689 baerlem5blem1 41691 hdmap14lem2a 41849 hdmap14lem4a 41853 hdmap14lem6 41855 hgmapval0 41874 selvvvval 42571 prjspersym 42593 prjspreln0 42595 prjspner1 42612 lmod0rng 48072 scmsuppss 48215 lmodvsmdi 48223 ply1mulgsumlem4 48234 lincval1 48264 lincvalsc0 48266 linc0scn0 48268 linc1 48270 ldepsprlem 48317 |
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