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Theorem slmd0vs 31477
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 29372 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 483 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑊 ∈ SLMod)
2 slmd0vs.f . . . . . 6 𝐹 = (Scalar‘𝑊)
3 eqid 2738 . . . . . 6 (Base‘𝐹) = (Base‘𝐹)
4 slmd0vs.o . . . . . 6 𝑂 = (0g𝐹)
52, 3, 4slmd0cl 31471 . . . . 5 (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹))
65adantr 481 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑂 ∈ (Base‘𝐹))
7 simpr 485 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑋𝑉)
8 slmd0vs.v . . . . 5 𝑉 = (Base‘𝑊)
9 eqid 2738 . . . . 5 (+g𝑊) = (+g𝑊)
10 slmd0vs.s . . . . 5 · = ( ·𝑠𝑊)
11 slmd0vs.z . . . . 5 0 = (0g𝑊)
12 eqid 2738 . . . . 5 (+g𝐹) = (+g𝐹)
13 eqid 2738 . . . . 5 (.r𝐹) = (.r𝐹)
14 eqid 2738 . . . . 5 (1r𝐹) = (1r𝐹)
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 31456 . . . 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 1378 . . 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 496 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))
1817simp3d 1143 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  .rcmulr 16963  Scalarcsca 16965   ·𝑠 cvsca 16966  0gc0g 17150  1rcur 19737  SLModcslmd 31453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-cmn 19388  df-srg 19742  df-slmd 31454
This theorem is referenced by:  slmdvs0  31478  gsumvsca2  31480
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