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Theorem slmd0vs 32639
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 30530 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0vs.v 𝑉 = (Baseβ€˜π‘Š)
slmd0vs.f 𝐹 = (Scalarβ€˜π‘Š)
slmd0vs.s Β· = ( ·𝑠 β€˜π‘Š)
slmd0vs.o 𝑂 = (0gβ€˜πΉ)
slmd0vs.z 0 = (0gβ€˜π‘Š)
Assertion
Ref Expression
slmd0vs ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑂 Β· 𝑋) = 0 )
Proof of Theorem slmd0vs
StepHypRef Expression
1 simpl 481 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ π‘Š ∈ SLMod)
2 slmd0vs.f . . . . . 6 𝐹 = (Scalarβ€˜π‘Š)
3 eqid 2730 . . . . . 6 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
4 slmd0vs.o . . . . . 6 𝑂 = (0gβ€˜πΉ)
52, 3, 4slmd0cl 32633 . . . . 5 (π‘Š ∈ SLMod β†’ 𝑂 ∈ (Baseβ€˜πΉ))
65adantr 479 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑂 ∈ (Baseβ€˜πΉ))
7 simpr 483 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ 𝑉)
8 slmd0vs.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
9 eqid 2730 . . . . 5 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
10 slmd0vs.s . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
11 slmd0vs.z . . . . 5 0 = (0gβ€˜π‘Š)
12 eqid 2730 . . . . 5 (+gβ€˜πΉ) = (+gβ€˜πΉ)
13 eqid 2730 . . . . 5 (.rβ€˜πΉ) = (.rβ€˜πΉ)
14 eqid 2730 . . . . 5 (1rβ€˜πΉ) = (1rβ€˜πΉ)
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 32618 . . . 4 ((π‘Š ∈ SLMod ∧ (𝑂 ∈ (Baseβ€˜πΉ) ∧ 𝑂 ∈ (Baseβ€˜πΉ)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (((𝑂 Β· 𝑋) ∈ 𝑉 ∧ (𝑂 Β· (𝑋(+gβ€˜π‘Š)𝑋)) = ((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋)) ∧ ((𝑂(+gβ€˜πΉ)𝑂) Β· 𝑋) = ((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋))) ∧ (((𝑂(.rβ€˜πΉ)𝑂) Β· 𝑋) = (𝑂 Β· (𝑂 Β· 𝑋)) ∧ ((1rβ€˜πΉ) Β· 𝑋) = 𝑋 ∧ (𝑂 Β· 𝑋) = 0 )))
161, 6, 6, 7, 7, 15syl122anc 1377 . . 3 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ (((𝑂 Β· 𝑋) ∈ 𝑉 ∧ (𝑂 Β· (𝑋(+gβ€˜π‘Š)𝑋)) = ((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋)) ∧ ((𝑂(+gβ€˜πΉ)𝑂) Β· 𝑋) = ((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋))) ∧ (((𝑂(.rβ€˜πΉ)𝑂) Β· 𝑋) = (𝑂 Β· (𝑂 Β· 𝑋)) ∧ ((1rβ€˜πΉ) Β· 𝑋) = 𝑋 ∧ (𝑂 Β· 𝑋) = 0 )))
1716simprd 494 . 2 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ (((𝑂(.rβ€˜πΉ)𝑂) Β· 𝑋) = (𝑂 Β· (𝑂 Β· 𝑋)) ∧ ((1rβ€˜πΉ) Β· 𝑋) = 𝑋 ∧ (𝑂 Β· 𝑋) = 0 ))
1817simp3d 1142 1 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑂 Β· 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389  1rcur 20075  SLModcslmd 32615
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7367  df-ov 7414  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-cmn 19691  df-srg 20081  df-slmd 32616
This theorem is referenced by:  slmdvs0  32640  gsumvsca2  32642
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