Theorem slmd0vs 33406

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31215 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 486	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ SLMod)
2		slmd0vs.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2764	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
4		slmd0vs.o	. . . . . 6 ⊢ 𝑂 = (0g‘𝐹)
5	2, 3, 4	slmd0cl 33400	. . . . 5 ⊢ (𝑊 ∈ SLMod → 𝑂 ∈ (Base‘𝐹))
6	5	adantr 484	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑂 ∈ (Base‘𝐹))
7		simpr 488	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
8		slmd0vs.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
9		eqid 2764	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
10		slmd0vs.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
11		slmd0vs.z	. . . . 5 ⊢ 0 = (0g‘𝑊)
12		eqid 2764	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
13		eqid 2764	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
14		eqid 2764	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
15	8, 9, 10, 11, 2, 3, 12, 13, 14, 4	slmdlema 33385	. . . 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 1400	. . 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 499	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (((𝑂(.r‘𝐹)𝑂) · 𝑋) = (𝑂 · (𝑂 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ (𝑂 · 𝑋) = 0 ))
18	17	simp3d 1158	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 )

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  ._rcmulr 17289  Scalarcsca 17291   ·_𝑠 cvsca 17292  0_gc0g 17470  1_rcur 20233  SLModcslmd 33382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-riota 7355  df-ov 7401  df-0g 17472  df-mgm 18676  df-sgrp 18755  df-mnd 18771  df-cmn 19824  df-srg 20239  df-slmd 33383

This theorem is referenced by:  slmdvs0  33407  gsumvsca2  33409