Theorem slmd0vs 33183

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 30945 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 2730	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
4		slmd0vs.o	. . . . . 6 ⊢ 𝑂 = (0g‘𝐹)
5	2, 3, 4	slmd0cl 33177	. . . . 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 2730	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
10		slmd0vs.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
11		slmd0vs.z	. . . . 5 ⊢ 0 = (0g‘𝑊)
12		eqid 2730	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
13		eqid 2730	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
14		eqid 2730	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
15	8, 9, 10, 11, 2, 3, 12, 13, 14, 4	slmdlema 33162	. . . 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 1381	. . 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 1144	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  +_gcplusg 17226  ._rcmulr 17227  Scalarcsca 17229   ·_𝑠 cvsca 17230  0_gc0g 17408  1_rcur 20096  SLModcslmd 33159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-riota 7346  df-ov 7392  df-0g 17410  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-cmn 19718  df-srg 20102  df-slmd 33160

This theorem is referenced by:  slmdvs0  33184  gsumvsca2  33186