Theorem slmdvs0 30323

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28436 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
slmdvs0.f	⊢ 𝐹 = (Scalar‘𝑊)
slmdvs0.s	⊢ · = ( ·𝑠 ‘𝑊)
slmdvs0.k	⊢ 𝐾 = (Base‘𝐹)
slmdvs0.z	⊢ 0 = (0g‘𝑊)

Assertion

Ref	Expression
slmdvs0	⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )

Proof of Theorem slmdvs0

Step	Hyp	Ref	Expression
1		slmdvs0.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	slmdsrg 30305	. . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3		slmdvs0.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
4		eqid 2825	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
5		eqid 2825	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
6	3, 4, 5	srgrz 18880	. . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
7	2, 6	sylan 577	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
8	7	oveq1d 6920	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 ))
9		simpl 476	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod)
10		simpr 479	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
11	2	adantr 474	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing)
12	3, 5	srg0cl 18873	. . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾)
13	11, 12	syl 17	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾)
14		eqid 2825	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
15		slmdvs0.z	. . . . . 6 ⊢ 0 = (0g‘𝑊)
16	14, 15	slmd0vcl 30319	. . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
17	16	adantr 474	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊))
18		slmdvs0.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
19	14, 1, 18, 3, 4	slmdvsass 30315	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ (𝑋 ∈ 𝐾 ∧ (0g‘𝐹) ∈ 𝐾 ∧ 0 ∈ (Base‘𝑊))) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 )))
20	9, 10, 13, 17, 19	syl13anc 1497	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 )))
21	14, 1, 18, 5, 15	slmd0vs 30322	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 )
22	17, 21	syldan 587	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 )
23	22	oveq2d 6921	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 ))
24	20, 23	eqtrd 2861	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 ))
25	8, 24, 22	3eqtr3d 2869	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  ._rcmulr 16306  Scalarcsca 16308   ·_𝑠 cvsca 16309  0_gc0g 16453  SRingcsrg 18859  SLModcslmd 30298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-riota 6866  df-ov 6908  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-cmn 18548  df-srg 18860  df-slmd 30299

This theorem is referenced by:  gsumvsca1  30327