Theorem slmdvs0 33178

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 30953 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 33160	. . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3		slmdvs0.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
4		eqid 2729	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
5		eqid 2729	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
6	3, 4, 5	srgrz 20116	. . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
7	2, 6	sylan 580	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
8	7	oveq1d 7402	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 ))
9		simpl 482	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod)
10		simpr 484	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
11	2	adantr 480	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing)
12	3, 5	srg0cl 20109	. . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾)
13	11, 12	syl 17	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾)
14		eqid 2729	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
15		slmdvs0.z	. . . . . 6 ⊢ 0 = (0g‘𝑊)
16	14, 15	slmd0vcl 33174	. . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
17	16	adantr 480	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊))
18		slmdvs0.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
19	14, 1, 18, 3, 4	slmdvsass 33170	. . . 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 1374	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 )))
21	14, 1, 18, 5, 15	slmd0vs 33177	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 )
22	17, 21	syldan 591	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 )
23	22	oveq2d 7403	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 ))
24	20, 23	eqtrd 2764	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 ))
25	8, 24, 22	3eqtr3d 2772	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402  SRingcsrg 20095  SLModcslmd 33153

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-riota 7344  df-ov 7390  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-cmn 19712  df-srg 20096  df-slmd 33154

This theorem is referenced by:  gsumvsca1  33179