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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 28787 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 485 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ LMod) | |
2 | lmod0vs.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 19642 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Ring) |
5 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | lmod0vs.o | . . . . . . 7 ⊢ 𝑂 = (0g‘𝐹) | |
7 | 5, 6 | ring0cl 19319 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝑂 ∈ (Base‘𝐹)) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹)) |
9 | simpr 487 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
10 | lmod0vs.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
12 | lmod0vs.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
13 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
14 | 10, 11, 2, 12, 5, 13 | lmodvsdir 19658 | . . . . 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 1368 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) |
16 | ringgrp 19302 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
17 | 4, 16 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ Grp) |
18 | 5, 13, 6 | grplid 18133 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑂 ∈ (Base‘𝐹)) → (𝑂(+g‘𝐹)𝑂) = 𝑂) |
19 | 17, 8, 18 | syl2anc 586 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂(+g‘𝐹)𝑂) = 𝑂) |
20 | 19 | oveq1d 7171 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂(+g‘𝐹)𝑂) · 𝑋) = (𝑂 · 𝑋)) |
21 | 15, 20 | eqtr3d 2858 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋)) |
22 | 10, 2, 12, 5 | lmodvscl 19651 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑂 ∈ (Base‘𝐹) ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
23 | 1, 8, 9, 22 | syl3anc 1367 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) ∈ 𝑉) |
24 | lmod0vs.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
25 | 10, 11, 24 | lmod0vid 19666 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑂 · 𝑋) ∈ 𝑉) → (((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋) ↔ 0 = (𝑂 · 𝑋))) |
26 | 23, 25 | syldan 593 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) = (𝑂 · 𝑋) ↔ 0 = (𝑂 · 𝑋))) |
27 | 21, 26 | mpbid 234 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 0 = (𝑂 · 𝑋)) |
28 | 27 | eqcomd 2827 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 Grpcgrp 18103 Ringcrg 19297 LModclmod 19634 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-ring 19299 df-lmod 19636 |
This theorem is referenced by: lmodvs0 19668 lmodvsmmulgdi 19669 lcomfsupp 19674 lmodvneg1 19677 mptscmfsupp0 19699 lvecvs0or 19880 lssvs0or 19882 lspsneleq 19887 lspdisj 19897 lspfixed 19900 lspexch 19901 lspsolvlem 19914 lspsolv 19915 ascl0 20113 mplcoe1 20246 mplbas2 20251 ply10s0 20424 ply1scl0 20458 gsummoncoe1 20472 uvcresum 20937 frlmsslsp 20940 frlmup1 20942 frlmup2 20943 pmatcollpwscmatlem1 21397 idpm2idmp 21409 mp2pm2mplem4 21417 pm2mpmhmlem1 21426 monmat2matmon 21432 cpmidpmatlem3 21480 clm0vs 23699 plypf1 24802 lmodslmd 30832 lbsdiflsp0 31022 fedgmullem2 31026 lshpkrlem1 36261 ldual0vs 36311 lclkrlem1 38657 lcd0vs 38766 baerlem3lem1 38858 baerlem5blem1 38860 hdmap14lem2a 39018 hdmap14lem4a 39022 hdmap14lem6 39024 hgmapval0 39043 prjspersym 39277 prjspreln0 39279 lmod0rng 44159 scmsuppss 44440 lmodvsmdi 44450 ply1mulgsumlem4 44463 lincval1 44494 lincvalsc0 44496 linc0scn0 44498 linc1 44500 ldepsprlem 44547 |
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