Theorem slmd0vs 33485

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31303 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 487	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ SLMod)
2		slmd0vs.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2769	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
4		slmd0vs.o	. . . . . 6 ⊢ 𝑂 = (0g‘𝐹)
5	2, 3, 4	slmd0cl 33479	. . . . 5 ⊢ (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹))
6	5	adantr 485	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹))
7		simpr 489	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
8		slmd0vs.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
9		eqid 2769	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
10		slmd0vs.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
11		slmd0vs.z	. . . . 5 ⊢ 0 = (0g‘𝑊)
12		eqid 2769	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
13		eqid 2769	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
14		eqid 2769	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
15	8, 9, 10, 11, 2, 3, 12, 13, 14, 4	slmdlema 33464	. . . 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 1404	. . 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 500	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))
18	17	simp3d 1160	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 )

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  ._rcmulr 17311  Scalarcsca 17313   ·_𝑠 cvsca 17314  0_gc0g 17492  1_rcur 20263  SLModcslmd 33461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-riota 7368  df-ov 7414  df-0g 17494  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-cmn 19852  df-srg 20269  df-slmd 33462

This theorem is referenced by:  slmdvs0  33486  gsumvsca2  33488