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Theorem slmdvs0 28902
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 27058 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 28884 . . . 4 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmdvs0.k . . . . 5 𝐾 = (Base‘𝐹)
4 eqid 2609 . . . . 5 (.r𝐹) = (.r𝐹)
5 eqid 2609 . . . . 5 (0g𝐹) = (0g𝐹)
63, 4, 5srgrz 18297 . . . 4 ((𝐹 ∈ SRing ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
72, 6sylan 486 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
87oveq1d 6541 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = ((0g𝐹) · 0 ))
9 simpl 471 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑊 ∈ SLMod)
10 simpr 475 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑋𝐾)
112adantr 479 . . . . 5 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝐹 ∈ SRing)
123, 5srg0cl 18290 . . . . 5 (𝐹 ∈ SRing → (0g𝐹) ∈ 𝐾)
1311, 12syl 17 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (0g𝐹) ∈ 𝐾)
14 eqid 2609 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
15 slmdvs0.z . . . . . 6 0 = (0g𝑊)
1614, 15slmd0vcl 28898 . . . . 5 (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
1716adantr 479 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 0 ∈ (Base‘𝑊))
18 slmdvs0.s . . . . 5 · = ( ·𝑠𝑊)
1914, 1, 18, 3, 4slmdvsass 28894 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑋𝐾 ∧ (0g𝐹) ∈ 𝐾0 ∈ (Base‘𝑊))) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
209, 10, 13, 17, 19syl13anc 1319 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
2114, 1, 18, 5, 15slmd0vs 28901 . . . . 5 ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g𝐹) · 0 ) = 0 )
2217, 21syldan 485 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((0g𝐹) · 0 ) = 0 )
2322oveq2d 6542 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · ((0g𝐹) · 0 )) = (𝑋 · 0 ))
2420, 23eqtrd 2643 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · 0 ))
258, 24, 223eqtr3d 2651 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cfv 5789  (class class class)co 6526  Basecbs 15643  .rcmulr 15717  Scalarcsca 15719   ·𝑠 cvsca 15720  0gc0g 15871  SRingcsrg 18276  SLModcslmd 28877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-riota 6488  df-ov 6529  df-0g 15873  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-cmn 17966  df-srg 18277  df-slmd 28878
This theorem is referenced by:  gsumvsca1  28906
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