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Mirrors > Home > MPE Home > Th. List > lmodvs0 | Structured version Visualization version GIF version |
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28729 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvs0.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvs0.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvs0.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodvs0.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lmodvs0 | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvs0.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | lmodring 19573 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
3 | lmodvs0.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
4 | eqid 2821 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
5 | eqid 2821 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 3, 4, 5 | ringrz 19269 | . . . 4 ⊢ ((𝐹 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) |
7 | 2, 6 | sylan 580 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹)) |
8 | 7 | oveq1d 7160 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 )) |
9 | simpl 483 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ LMod) | |
10 | simpr 485 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
11 | 2 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ Ring) |
12 | 3, 5 | ring0cl 19250 | . . . . 5 ⊢ (𝐹 ∈ Ring → (0g‘𝐹) ∈ 𝐾) |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾) |
14 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
15 | lmodvs0.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
16 | 14, 15 | lmod0vcl 19594 | . . . . 5 ⊢ (𝑊 ∈ LMod → 0 ∈ (Base‘𝑊)) |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊)) |
18 | lmodvs0.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
19 | 14, 1, 18, 3, 4 | lmodvsass 19590 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝐾 ∧ (0g‘𝐹) ∈ 𝐾 ∧ 0 ∈ (Base‘𝑊))) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 ))) |
20 | 9, 10, 13, 17, 19 | syl13anc 1364 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 ))) |
21 | 14, 1, 18, 5, 15 | lmod0vs 19598 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 ) |
22 | 17, 21 | syldan 591 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 ) |
23 | 22 | oveq2d 7161 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 )) |
24 | 20, 23 | eqtrd 2856 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 )) |
25 | 8, 24, 22 | 3eqtr3d 2864 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 .rcmulr 16556 Scalarcsca 16558 ·𝑠 cvsca 16559 0gc0g 16703 Ringcrg 19228 LModclmod 19565 |
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 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-plusg 16568 df-0g 16705 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-grp 18046 df-mgp 19171 df-ring 19230 df-lmod 19567 |
This theorem is referenced by: lmodfopne 19603 lsssn0 19650 lmodvsinv2 19740 0lmhm 19743 lvecvs0or 19811 dsmmlss 20818 pmatcollpwfi 21320 pmatcollpw3fi1lem1 21324 pm2mp 21363 chfacfscmul0 21396 ttgbtwnid 26598 0nellinds 30863 lcdvs0N 38634 hdmap14lem13 38898 lmodvsmdi 44328 linc0scn0 44376 |
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