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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 31029 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 20866 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
| 4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Ring) |
| 5 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
| 6 | lmod0vs.o | . . . . . . 7 ⊢ 𝑂 = (0g‘𝐹) | |
| 7 | 5, 6 | ring0cl 20264 | . . . . . 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 2737 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 12 | lmod0vs.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 13 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
| 14 | 10, 11, 2, 12, 5, 13 | lmodvsdir 20884 | . . . . 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 1374 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) |
| 16 | ringgrp 20235 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
| 17 | 4, 16 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Grp) |
| 18 | 5, 13, 6 | grplid 18985 | . . . . . 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 7446 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = (𝑂 · 𝑋)) |
| 21 | 15, 20 | eqtr3d 2779 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋)) |
| 22 | 10, 2, 12, 5 | lmodvscl 20876 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑂 ∈ (Base‘𝐹) ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
| 23 | 1, 8, 9, 22 | syl3anc 1373 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
| 24 | lmod0vs.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
| 25 | 10, 11, 24 | lmod0vid 20892 | . . . 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 2743 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 Grpcgrp 18951 Ringcrg 20230 LModclmod 20858 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ring 20232 df-lmod 20860 |
| This theorem is referenced by: lmodvs0 20894 lmodvsmmulgdi 20895 lcomfsupp 20900 lmodvneg1 20903 mptscmfsupp0 20925 lvecvs0or 21110 lssvs0or 21112 lspsneleq 21117 lspdisj 21127 lspfixed 21130 lspexch 21131 lspsolvlem 21144 lspsolv 21145 uvcresum 21813 frlmsslsp 21816 frlmup1 21818 frlmup2 21819 ascl0 21904 mplcoe1 22055 mplbas2 22060 ply10s0 22259 ply1scl0OLD 22294 gsummoncoe1 22312 evls1fpws 22373 pmatcollpwscmatlem1 22795 idpm2idmp 22807 mp2pm2mplem4 22815 pm2mpmhmlem1 22824 monmat2matmon 22830 cpmidpmatlem3 22878 clm0vs 25128 plypf1 26251 lmodslmd 33210 r1p0 33626 ply1degltdimlem 33673 lbsdiflsp0 33677 fedgmullem2 33681 lshpkrlem1 39111 ldual0vs 39161 lclkrlem1 41508 lcd0vs 41617 baerlem3lem1 41709 baerlem5blem1 41711 hdmap14lem2a 41869 hdmap14lem4a 41873 hdmap14lem6 41875 hgmapval0 41894 selvvvval 42595 prjspersym 42617 prjspreln0 42619 prjspner1 42636 lmod0rng 48145 scmsuppss 48287 lmodvsmdi 48295 ply1mulgsumlem4 48306 lincval1 48336 lincvalsc0 48338 linc0scn0 48340 linc1 48342 ldepsprlem 48389 |
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