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Theorem slmd0vs 28902
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27045 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 472 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑊 ∈ SLMod)
2 slmd0vs.f . . . . . 6 𝐹 = (Scalar‘𝑊)
3 eqid 2610 . . . . . 6 (Base‘𝐹) = (Base‘𝐹)
4 slmd0vs.o . . . . . 6 𝑂 = (0g𝐹)
52, 3, 4slmd0cl 28896 . . . . 5 (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹))
65adantr 480 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑂 ∈ (Base‘𝐹))
7 simpr 476 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑋𝑉)
8 slmd0vs.v . . . . 5 𝑉 = (Base‘𝑊)
9 eqid 2610 . . . . 5 (+g𝑊) = (+g𝑊)
10 slmd0vs.s . . . . 5 · = ( ·𝑠𝑊)
11 slmd0vs.z . . . . 5 0 = (0g𝑊)
12 eqid 2610 . . . . 5 (+g𝐹) = (+g𝐹)
13 eqid 2610 . . . . 5 (.r𝐹) = (.r𝐹)
14 eqid 2610 . . . . 5 (1r𝐹) = (1r𝐹)
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 28881 . . . 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 1327 . . 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 478 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))
1817simp3d 1068 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  cfv 5790  (class class class)co 6527  Basecbs 15644  +gcplusg 15717  .rcmulr 15718  Scalarcsca 15720   ·𝑠 cvsca 15721  0gc0g 15872  1rcur 18273  SLModcslmd 28878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-riota 6489  df-ov 6530  df-0g 15874  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-cmn 17967  df-srg 18278  df-slmd 28879
This theorem is referenced by:  slmdvs0  28903  gsumvsca2  28908
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