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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 28793 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 486 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ LMod) | |
2 | lmod0vs.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 19635 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Ring) |
5 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | lmod0vs.o | . . . . . . 7 ⊢ 𝑂 = (0g‘𝐹) | |
7 | 5, 6 | ring0cl 19315 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝑂 ∈ (Base‘𝐹)) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹)) |
9 | simpr 488 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
10 | lmod0vs.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | eqid 2798 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
12 | lmod0vs.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
13 | eqid 2798 | . . . . . 6 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
14 | 10, 11, 2, 12, 5, 13 | lmodvsdir 19651 | . . . . 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 1369 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) |
16 | ringgrp 19295 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
17 | 4, 16 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Grp) |
18 | 5, 13, 6 | grplid 18125 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑂 ∈ (Base‘𝐹)) → (𝑂(+g‘𝐹)𝑂) = 𝑂) |
19 | 17, 8, 18 | syl2anc 587 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂(+g‘𝐹)𝑂) = 𝑂) |
20 | 19 | oveq1d 7150 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = (𝑂 · 𝑋)) |
21 | 15, 20 | eqtr3d 2835 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋)) |
22 | 10, 2, 12, 5 | lmodvscl 19644 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑂 ∈ (Base‘𝐹) ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
23 | 1, 8, 9, 22 | syl3anc 1368 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
24 | lmod0vs.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
25 | 10, 11, 24 | lmod0vid 19659 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑂 · 𝑋) ∈ 𝑉) → (((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋) ↔ 0 = (𝑂 · 𝑋))) |
26 | 23, 25 | syldan 594 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋) ↔ 0 = (𝑂 · 𝑋))) |
27 | 21, 26 | mpbid 235 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 0 = (𝑂 · 𝑋)) |
28 | 27 | eqcomd 2804 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 Grpcgrp 18095 Ringcrg 19290 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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
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-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 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-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-ring 19292 df-lmod 19629 |
This theorem is referenced by: lmodvs0 19661 lmodvsmmulgdi 19662 lcomfsupp 19667 lmodvneg1 19670 mptscmfsupp0 19692 lvecvs0or 19873 lssvs0or 19875 lspsneleq 19880 lspdisj 19890 lspfixed 19893 lspexch 19894 lspsolvlem 19907 lspsolv 19908 uvcresum 20482 frlmsslsp 20485 frlmup1 20487 frlmup2 20488 ascl0 20570 mplcoe1 20705 mplbas2 20710 ply10s0 20885 ply1scl0 20919 gsummoncoe1 20933 pmatcollpwscmatlem1 21394 idpm2idmp 21406 mp2pm2mplem4 21414 pm2mpmhmlem1 21423 monmat2matmon 21429 cpmidpmatlem3 21477 clm0vs 23700 plypf1 24809 lmodslmd 30882 lbsdiflsp0 31110 fedgmullem2 31114 lshpkrlem1 36406 ldual0vs 36456 lclkrlem1 38802 lcd0vs 38911 baerlem3lem1 39003 baerlem5blem1 39005 hdmap14lem2a 39163 hdmap14lem4a 39167 hdmap14lem6 39169 hgmapval0 39188 prjspersym 39601 prjspreln0 39603 lmod0rng 44492 scmsuppss 44774 lmodvsmdi 44784 ply1mulgsumlem4 44797 lincval1 44828 lincvalsc0 44830 linc0scn0 44832 linc1 44834 ldepsprlem 44881 |
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