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Theorem slmdvs0 30994
 Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28896 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 30976 . . . 4 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmdvs0.k . . . . 5 𝐾 = (Base‘𝐹)
4 eqid 2759 . . . . 5 (.r𝐹) = (.r𝐹)
5 eqid 2759 . . . . 5 (0g𝐹) = (0g𝐹)
63, 4, 5srgrz 19334 . . . 4 ((𝐹 ∈ SRing ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
72, 6sylan 584 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
87oveq1d 7163 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = ((0g𝐹) · 0 ))
9 simpl 487 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑊 ∈ SLMod)
10 simpr 489 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝑋𝐾)
112adantr 485 . . . . 5 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 𝐹 ∈ SRing)
123, 5srg0cl 19327 . . . . 5 (𝐹 ∈ SRing → (0g𝐹) ∈ 𝐾)
1311, 12syl 17 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (0g𝐹) ∈ 𝐾)
14 eqid 2759 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
15 slmdvs0.z . . . . . 6 0 = (0g𝑊)
1614, 15slmd0vcl 30990 . . . . 5 (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
1716adantr 485 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → 0 ∈ (Base‘𝑊))
18 slmdvs0.s . . . . 5 · = ( ·𝑠𝑊)
1914, 1, 18, 3, 4slmdvsass 30986 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑋𝐾 ∧ (0g𝐹) ∈ 𝐾0 ∈ (Base‘𝑊))) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
209, 10, 13, 17, 19syl13anc 1370 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
2114, 1, 18, 5, 15slmd0vs 30993 . . . . 5 ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g𝐹) · 0 ) = 0 )
2217, 21syldan 595 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((0g𝐹) · 0 ) = 0 )
2322oveq2d 7164 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · ((0g𝐹) · 0 )) = (𝑋 · 0 ))
2420, 23eqtrd 2794 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · 0 ))
258, 24, 223eqtr3d 2802 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ‘cfv 6333  (class class class)co 7148  Basecbs 16531  .rcmulr 16614  Scalarcsca 16616   ·𝑠 cvsca 16617  0gc0g 16761  SRingcsrg 19313  SLModcslmd 30969 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-iota 6292  df-fun 6335  df-fv 6341  df-riota 7106  df-ov 7151  df-0g 16763  df-mgm 17908  df-sgrp 17957  df-mnd 17968  df-cmn 18965  df-srg 19314  df-slmd 30970 This theorem is referenced by:  gsumvsca1  30995
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