Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  slmdvs0 Structured version   Visualization version   GIF version

Theorem slmdvs0 29906
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28009 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 29888 . . . 4 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmdvs0.k . . . . 5 𝐾 = (Base‘𝐹)
4 eqid 2651 . . . . 5 (.r𝐹) = (.r𝐹)
5 eqid 2651 . . . . 5 (0g𝐹) = (0g𝐹)
63, 4, 5srgrz 18572 . . . 4 ((𝐹 ∈ SRing ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
72, 6sylan 487 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋(.r𝐹)(0g𝐹)) = (0g𝐹))
87oveq1d 6705 . 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 18565 . . . . 5 (𝐹 ∈ SRing → (0g𝐹) ∈ 𝐾)
1311, 12syl 17 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (0g𝐹) ∈ 𝐾)
14 eqid 2651 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
15 slmdvs0.z . . . . . 6 0 = (0g𝑊)
1614, 15slmd0vcl 29902 . . . . 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 29898 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑋𝐾 ∧ (0g𝐹) ∈ 𝐾0 ∈ (Base‘𝑊))) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
209, 10, 13, 17, 19syl13anc 1368 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · ((0g𝐹) · 0 )))
2114, 1, 18, 5, 15slmd0vs 29905 . . . . 5 ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g𝐹) · 0 ) = 0 )
2217, 21syldan 486 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((0g𝐹) · 0 ) = 0 )
2322oveq2d 6706 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · ((0g𝐹) · 0 )) = (𝑋 · 0 ))
2420, 23eqtrd 2685 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → ((𝑋(.r𝐹)(0g𝐹)) · 0 ) = (𝑋 · 0 ))
258, 24, 223eqtr3d 2693 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  SRingcsrg 18551  SLModcslmd 29881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-cmn 18241  df-srg 18552  df-slmd 29882
This theorem is referenced by:  gsumvsca1  29910
  Copyright terms: Public domain W3C validator