Theorem slmdvs0 33407

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 31229 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 33389	. . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3		slmdvs0.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
4		eqid 2764	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
5		eqid 2764	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
6	3, 4, 5	srgrz 20259	. . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
7	2, 6	sylan 589	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
8	7	oveq1d 7413	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 ))
9		simpl 486	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod)
10		simpr 488	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
11	2	adantr 484	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing)
12	3, 5	srg0cl 20252	. . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾)
13	11, 12	syl 17	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾)
14		eqid 2764	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
15		slmdvs0.z	. . . . . 6 ⊢ 0 = (0g‘𝑊)
16	14, 15	slmd0vcl 33403	. . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
17	16	adantr 484	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊))
18		slmdvs0.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
19	14, 1, 18, 3, 4	slmdvsass 33399	. . . 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 1393	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 )))
21	14, 1, 18, 5, 15	slmd0vs 33406	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 )
22	17, 21	syldan 600	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 )
23	22	oveq2d 7414	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 ))
24	20, 23	eqtrd 2799	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 ))
25	8, 24, 22	3eqtr3d 2807	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  ._rcmulr 17289  Scalarcsca 17291   ·_𝑠 cvsca 17292  0_gc0g 17470  SRingcsrg 20238  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:  gsumvsca1  33408