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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 29913 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 20328 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
3 | lmodvs0.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
4 | eqid 2736 | . . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
5 | eqid 2736 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 3, 4, 5 | ringrz 20010 | . . . 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 7371 | . 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 19988 | . . . . 5 ⊢ (𝐹 ∈ Ring → (0g‘𝐹) ∈ 𝐾) |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾) |
14 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
15 | lmodvs0.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
16 | 14, 15 | lmod0vcl 20349 | . . . . 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 20345 | . . . 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 1372 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 ))) |
21 | 14, 1, 18, 5, 15 | lmod0vs 20353 | . . . . 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 7372 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 )) |
24 | 20, 23 | eqtrd 2776 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 )) |
25 | 8, 24, 22 | 3eqtr3d 2784 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 .rcmulr 17133 Scalarcsca 17135 ·𝑠 cvsca 17136 0gc0g 17320 Ringcrg 19962 LModclmod 20320 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-plusg 17145 df-0g 17322 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-grp 18750 df-mgp 19895 df-ring 19964 df-lmod 20322 |
This theorem is referenced by: lmodfopne 20358 lsssn0 20406 lmodvsinv2 20496 0lmhm 20499 lvecvs0or 20567 dsmmlss 21148 pmatcollpwfi 22129 pmatcollpw3fi1lem1 22133 pm2mp 22172 chfacfscmul0 22205 ttgbtwnid 27779 0nellinds 32103 lcdvs0N 40069 hdmap14lem13 40333 lmodvsmdi 46429 linc0scn0 46475 |
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