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Theorem slmdvs0 32401
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 30308 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 = (Scalarβ€˜π‘Š)
slmdvs0.s Β· = ( ·𝑠 β€˜π‘Š)
slmdvs0.k 𝐾 = (Baseβ€˜πΉ)
slmdvs0.z 0 = (0gβ€˜π‘Š)
Assertion
Ref Expression
slmdvs0 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ (𝑋 Β· 0 ) = 0 )
Proof of Theorem slmdvs0
StepHypRef Expression
1 slmdvs0.f . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
21slmdsrg 32383 . . . 4 (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
3 slmdvs0.k . . . . 5 𝐾 = (Baseβ€˜πΉ)
4 eqid 2733 . . . . 5 (.rβ€˜πΉ) = (.rβ€˜πΉ)
5 eqid 2733 . . . . 5 (0gβ€˜πΉ) = (0gβ€˜πΉ)
63, 4, 5srgrz 20030 . . . 4 ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) β†’ (𝑋(.rβ€˜πΉ)(0gβ€˜πΉ)) = (0gβ€˜πΉ))
72, 6sylan 581 . . 3 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ (𝑋(.rβ€˜πΉ)(0gβ€˜πΉ)) = (0gβ€˜πΉ))
87oveq1d 7424 . 2 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ ((𝑋(.rβ€˜πΉ)(0gβ€˜πΉ)) Β· 0 ) = ((0gβ€˜πΉ) Β· 0 ))
9 simpl 484 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ π‘Š ∈ SLMod)
10 simpr 486 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ 𝑋 ∈ 𝐾)
112adantr 482 . . . . 5 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ 𝐹 ∈ SRing)
123, 5srg0cl 20023 . . . . 5 (𝐹 ∈ SRing β†’ (0gβ€˜πΉ) ∈ 𝐾)
1311, 12syl 17 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ (0gβ€˜πΉ) ∈ 𝐾)
14 eqid 2733 . . . . . 6 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
15 slmdvs0.z . . . . . 6 0 = (0gβ€˜π‘Š)
1614, 15slmd0vcl 32397 . . . . 5 (π‘Š ∈ SLMod β†’ 0 ∈ (Baseβ€˜π‘Š))
1716adantr 482 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ 0 ∈ (Baseβ€˜π‘Š))
18 slmdvs0.s . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
1914, 1, 18, 3, 4slmdvsass 32393 . . . 4 ((π‘Š ∈ SLMod ∧ (𝑋 ∈ 𝐾 ∧ (0gβ€˜πΉ) ∈ 𝐾 ∧ 0 ∈ (Baseβ€˜π‘Š))) β†’ ((𝑋(.rβ€˜πΉ)(0gβ€˜πΉ)) Β· 0 ) = (𝑋 Β· ((0gβ€˜πΉ) Β· 0 )))
209, 10, 13, 17, 19syl13anc 1373 . . 3 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ ((𝑋(.rβ€˜πΉ)(0gβ€˜πΉ)) Β· 0 ) = (𝑋 Β· ((0gβ€˜πΉ) Β· 0 )))
2114, 1, 18, 5, 15slmd0vs 32400 . . . . 5 ((π‘Š ∈ SLMod ∧ 0 ∈ (Baseβ€˜π‘Š)) β†’ ((0gβ€˜πΉ) Β· 0 ) = 0 )
2217, 21syldan 592 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ ((0gβ€˜πΉ) Β· 0 ) = 0 )
2322oveq2d 7425 . . 3 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ (𝑋 Β· ((0gβ€˜πΉ) Β· 0 )) = (𝑋 Β· 0 ))
2420, 23eqtrd 2773 . 2 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ ((𝑋(.rβ€˜πΉ)(0gβ€˜πΉ)) Β· 0 ) = (𝑋 Β· 0 ))
258, 24, 223eqtr3d 2781 1 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ (𝑋 Β· 0 ) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  SRingcsrg 20009  SLModcslmd 32376
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-cmn 19650  df-srg 20010  df-slmd 32377
This theorem is referenced by:  gsumvsca1  32402
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