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Theorem slmdvs0 29109
 Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 27265 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvs0.f 𝐹 = (Scalar‘𝑊)
slmdvs0.s · = ( ·𝑠𝑊)
slmdvs0.k 𝐾 = (Base‘𝐹)
slmdvs0.z 0 = (0g𝑊)
Assertion
Ref Expression
slmdvs0 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )

Proof of Theorem slmdvs0
StepHypRef Expression
1 slmdvs0.f . . . . 5 𝐹 = (Scalar‘𝑊)
21slmdsrg 29091 . . . 4 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmdvs0.k . . . . 5 𝐾 = (Base‘𝐹)
4 eqid 2610 . . . . 5 (.r𝐹) = (.r𝐹)
5 eqid 2610 . . . . 5 (0g𝐹) = (0g𝐹)
63, 4, 5srgrz 18349 . . . 4 ((𝐹 ∈ SRing ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
72, 6sylan 487 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
87oveq1d 6564 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = ((0g𝐹) · 0 ))
9 simpl 472 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑊 ∈ SLMod)
10 simpr 476 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑋𝐾)
112adantr 480 . . . . 5 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝐹 ∈ SRing)
123, 5srg0cl 18342 . . . . 5 (𝐹 ∈ SRing → (0g𝐹) ∈ 𝐾)
1311, 12syl 17 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (0g𝐹) ∈ 𝐾)
14 eqid 2610 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
15 slmdvs0.z . . . . . 6 0 = (0g𝑊)
1614, 15slmd0vcl 29105 . . . . 5 (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
1716adantr 480 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 0 ∈ (Base‘𝑊))
18 slmdvs0.s . . . . 5 · = ( ·𝑠𝑊)
1914, 1, 18, 3, 4slmdvsass 29101 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑋𝐾 ∧ (0g𝐹) ∈ 𝐾0 ∈ (Base‘𝑊))) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
209, 10, 13, 17, 19syl13anc 1320 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
2114, 1, 18, 5, 15slmd0vs 29108 . . . . 5 ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g𝐹) · 0 ) = 0 )
2217, 21syldan 486 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((0g𝐹) · 0 ) = 0 )
2322oveq2d 6565 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · ((0g𝐹) · 0 )) = (𝑋 · 0 ))
2420, 23eqtrd 2644 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · 0 ))
258, 24, 223eqtr3d 2652 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  SRingcsrg 18328  SLModcslmd 29084 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 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 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 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-cmn 18018  df-srg 18329  df-slmd 29085 This theorem is referenced by:  gsumvsca1  29113
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