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Theorem slmdvs0 33231
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 31043 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 33213 . . . 4 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmdvs0.k . . . . 5 𝐾 = (Base‘𝐹)
4 eqid 2737 . . . . 5 (.r𝐹) = (.r𝐹)
5 eqid 2737 . . . . 5 (0g𝐹) = (0g𝐹)
63, 4, 5srgrz 20204 . . . 4 ((𝐹 ∈ SRing ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
72, 6sylan 580 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
87oveq1d 7446 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = ((0g𝐹) · 0 ))
9 simpl 482 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑊 ∈ SLMod)
10 simpr 484 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑋𝐾)
112adantr 480 . . . . 5 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝐹 ∈ SRing)
123, 5srg0cl 20197 . . . . 5 (𝐹 ∈ SRing → (0g𝐹) ∈ 𝐾)
1311, 12syl 17 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (0g𝐹) ∈ 𝐾)
14 eqid 2737 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
15 slmdvs0.z . . . . . 6 0 = (0g𝑊)
1614, 15slmd0vcl 33227 . . . . 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 33223 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑋𝐾 ∧ (0g𝐹) ∈ 𝐾0 ∈ (Base‘𝑊))) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
209, 10, 13, 17, 19syl13anc 1374 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
2114, 1, 18, 5, 15slmd0vs 33230 . . . . 5 ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g𝐹) · 0 ) = 0 )
2217, 21syldan 591 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((0g𝐹) · 0 ) = 0 )
2322oveq2d 7447 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · ((0g𝐹) · 0 )) = (𝑋 · 0 ))
2420, 23eqtrd 2777 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · 0 ))
258, 24, 223eqtr3d 2785 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  SRingcsrg 20183  SLModcslmd 33206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-cmn 19800  df-srg 20184  df-slmd 33207
This theorem is referenced by:  gsumvsca1  33232
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