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Theorem slmd0vs 31054
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28945 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 = (Base‘𝑊)
slmd0vs.f 𝐹 = (Scalar‘𝑊)
slmd0vs.s · = ( ·𝑠𝑊)
slmd0vs.o 𝑂 = (0g𝐹)
slmd0vs.z 0 = (0g𝑊)
Assertion
Ref Expression
slmd0vs ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )

Proof of Theorem slmd0vs
StepHypRef Expression
1 simpl 486 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑊 ∈ SLMod)
2 slmd0vs.f . . . . . 6 𝐹 = (Scalar‘𝑊)
3 eqid 2738 . . . . . 6 (Base‘𝐹) = (Base‘𝐹)
4 slmd0vs.o . . . . . 6 𝑂 = (0g𝐹)
52, 3, 4slmd0cl 31048 . . . . 5 (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹))
65adantr 484 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑂 ∈ (Base‘𝐹))
7 simpr 488 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → 𝑋𝑉)
8 slmd0vs.v . . . . 5 𝑉 = (Base‘𝑊)
9 eqid 2738 . . . . 5 (+g𝑊) = (+g𝑊)
10 slmd0vs.s . . . . 5 · = ( ·𝑠𝑊)
11 slmd0vs.z . . . . 5 0 = (0g𝑊)
12 eqid 2738 . . . . 5 (+g𝐹) = (+g𝐹)
13 eqid 2738 . . . . 5 (.r𝐹) = (.r𝐹)
14 eqid 2738 . . . . 5 (1r𝐹) = (1r𝐹)
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 31033 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑂 ∈ (Base‘𝐹) ∧ 𝑂 ∈ (Base‘𝐹)) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g𝑊)𝑋)) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )))
161, 6, 6, 7, 7, 15syl122anc 1380 . . 3 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g𝑊)𝑋)) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )))
1716simprd 499 . 2 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (((𝑂(.r𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))
1817simp3d 1145 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  cfv 6339  (class class class)co 7170  Basecbs 16586  +gcplusg 16668  .rcmulr 16669  Scalarcsca 16671   ·𝑠 cvsca 16672  0gc0g 16816  1rcur 19370  SLModcslmd 31030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-riota 7127  df-ov 7173  df-0g 16818  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-cmn 19026  df-srg 19375  df-slmd 31031
This theorem is referenced by:  slmdvs0  31055  gsumvsca2  31057
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