Theorem slmd0vs 33245

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31055 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	⊢ · = ( ·𝑠 ‘𝑊)
slmd0vs.o	⊢ 𝑂 = (0g‘𝐹)
slmd0vs.z	⊢ 0 = (0g‘𝑊)

Assertion

Ref	Expression
slmd0vs	⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 )

Proof of Theorem slmd0vs

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ SLMod)
2		slmd0vs.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2737	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
4		slmd0vs.o	. . . . . 6 ⊢ 𝑂 = (0g‘𝐹)
5	2, 3, 4	slmd0cl 33239	. . . . 5 ⊢ (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹))
6	5	adantr 480	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹))
7		simpr 484	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
8		slmd0vs.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
9		eqid 2737	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
10		slmd0vs.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
11		slmd0vs.z	. . . . 5 ⊢ 0 = (0g‘𝑊)
12		eqid 2737	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
13		eqid 2737	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
14		eqid 2737	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
15	8, 9, 10, 11, 2, 3, 12, 13, 14, 4	slmdlema 33224	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ (𝑂 ∈ (Base‘𝐹) ∧ 𝑂 ∈ (Base‘𝐹)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g‘𝑊)𝑋)) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )))
16	1, 6, 6, 7, 7, 15	syl122anc 1380	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂 · 𝑋) ∈ 𝑉 ∧ (𝑂 · (𝑋(+g‘𝑊)𝑋)) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋)) ∧ ((𝑂(+g‘𝐹)𝑂) · 𝑋) = ((𝑂 · 𝑋)(+g‘𝑊)(𝑂 · 𝑋))) ∧ (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 )))
17	16	simprd 495	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))
18	17	simp3d 1145	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  +_gcplusg 17307  ._rcmulr 17308  Scalarcsca 17310   ·_𝑠 cvsca 17311  0_gc0g 17495  1_rcur 20208  SLModcslmd 33221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-iota 6522  df-fun 6571  df-fv 6577  df-riota 7395  df-ov 7441  df-0g 17497  df-mgm 18675  df-sgrp 18754  df-mnd 18770  df-cmn 19824  df-srg 20214  df-slmd 33222

This theorem is referenced by:  slmdvs0  33246  gsumvsca2  33248