Theorem slmdvs0 33787

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 31627 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 33769	. . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3		slmdvs0.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
4		eqid 2761	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
5		eqid 2761	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
6	3, 4, 5	srgrz 20434	. . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
7	2, 6	sylan 592	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
8	7	oveq1d 7436	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 ))
9		simpl 488	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod)
10		simpr 490	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
11	2	adantr 486	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing)
12	3, 5	srg0cl 20427	. . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾)
13	11, 12	syl 18	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾)
14		eqid 2761	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
15		slmdvs0.z	. . . . . 6 ⊢ 0 = (0g‘𝑊)
16	14, 15	slmd0vcl 33783	. . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
17	16	adantr 486	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊))
18		slmdvs0.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
19	14, 1, 18, 3, 4	slmdvsass 33779	. . . 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 1399	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 )))
21	14, 1, 18, 5, 15	slmd0vs 33786	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 )
22	17, 21	syldan 603	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 )
23	22	oveq2d 7437	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 ))
24	20, 23	eqtrd 2796	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 ))
25	8, 24, 22	3eqtr3d 2804	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ‘cfv 6539  (class class class)co 7421  Basecbs 17388  ._rcmulr 17430  Scalarcsca 17432   ·_𝑠 cvsca 17433  0_gc0g 17611  SRingcsrg 20413  SLModcslmd 33762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6495  df-fun 6541  df-fv 6547  df-riota 7378  df-ov 7424  df-0g 17613  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-cmn 19997  df-srg 20414  df-slmd 33763

This theorem is used by:  gsumvsca1  33788