Theorem slmd0vs 33285

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31066 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 2735	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
4		slmd0vs.o	. . . . . 6 ⊢ 𝑂 = (0g‘𝐹)
5	2, 3, 4	slmd0cl 33279	. . . . 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 2735	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
10		slmd0vs.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
11		slmd0vs.z	. . . . 5 ⊢ 0 = (0g‘𝑊)
12		eqid 2735	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
13		eqid 2735	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
14		eqid 2735	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
15	8, 9, 10, 11, 2, 3, 12, 13, 14, 4	slmdlema 33264	. . . 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 1382	. . 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 1542   ∈ wcel 2114  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  +_gcplusg 17179  ._rcmulr 17180  Scalarcsca 17182   ·_𝑠 cvsca 17183  0_gc0g 17361  1_rcur 20118  SLModcslmd 33261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6447  df-fun 6493  df-fv 6499  df-riota 7315  df-ov 7361  df-0g 17363  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-cmn 19713  df-srg 20124  df-slmd 33262

This theorem is referenced by:  slmdvs0  33286  gsumvsca2  33288