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Theorem slmd0vs 30107
 Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28197 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0vs.v 𝑉 = (Base‘𝑊)
slmd0vs.f 𝐹 = (Scalar‘𝑊)
slmd0vs.s · = ( ·𝑠𝑊)
slmd0vs.o 𝑂 = (0g𝐹)
slmd0vs.z 0 = (0g𝑊)
Assertion
Ref Expression
slmd0vs ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )

Proof of Theorem slmd0vs
StepHypRef Expression
1 simpl 474 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑊 ∈ SLMod)
2 slmd0vs.f . . . . . 6 𝐹 = (Scalar‘𝑊)
3 eqid 2760 . . . . . 6 (Base‘𝐹) = (Base‘𝐹)
4 slmd0vs.o . . . . . 6 𝑂 = (0g𝐹)
52, 3, 4slmd0cl 30101 . . . . 5 (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹))
65adantr 472 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑂 ∈ (Base‘𝐹))
7 simpr 479 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑋𝑉)
8 slmd0vs.v . . . . 5 𝑉 = (Base‘𝑊)
9 eqid 2760 . . . . 5 (+g𝑊) = (+g𝑊)
10 slmd0vs.s . . . . 5 · = ( ·𝑠𝑊)
11 slmd0vs.z . . . . 5 0 = (0g𝑊)
12 eqid 2760 . . . . 5 (+g𝐹) = (+g𝐹)
13 eqid 2760 . . . . 5 (.r𝐹) = (.r𝐹)
14 eqid 2760 . . . . 5 (1r𝐹) = (1r𝐹)
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 30086 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑂 ∈ (Base‘𝐹) ∧ 𝑂 ∈ (Base‘𝐹)) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g𝑊)𝑋)) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )))
161, 6, 6, 7, 7, 15syl122anc 1486 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g𝑊)𝑋)) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )))
1716simprd 482 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))
1817simp3d 1139 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ‘cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  .rcmulr 16164  Scalarcsca 16166   ·𝑠 cvsca 16167  0gc0g 16322  1rcur 18721  SLModcslmd 30083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-riota 6775  df-ov 6817  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-cmn 18415  df-srg 18726  df-slmd 30084 This theorem is referenced by:  slmdvs0  30108  gsumvsca2  30113
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