Theorem slmdvs0 30994

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28896 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 30976	. . . 4 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3		slmdvs0.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
4		eqid 2759	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
5		eqid 2759	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
6	3, 4, 5	srgrz 19334	. . . 4 ⊢ ((𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
7	2, 6	sylan 584	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
8	7	oveq1d 7163	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = ((0g‘𝐹) · 0 ))
9		simpl 487	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ SLMod)
10		simpr 489	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
11	2	adantr 485	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ SRing)
12	3, 5	srg0cl 19327	. . . . 5 ⊢ (𝐹 ∈ SRing → (0g‘𝐹) ∈ 𝐾)
13	11, 12	syl 17	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (0g‘𝐹) ∈ 𝐾)
14		eqid 2759	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
15		slmdvs0.z	. . . . . 6 ⊢ 0 = (0g‘𝑊)
16	14, 15	slmd0vcl 30990	. . . . 5 ⊢ (𝑊 ∈ SLMod → 0 ∈ (Base‘𝑊))
17	16	adantr 485	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → 0 ∈ (Base‘𝑊))
18		slmdvs0.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
19	14, 1, 18, 3, 4	slmdvsass 30986	. . . 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 1370	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · ((0g‘𝐹) · 0 )))
21	14, 1, 18, 5, 15	slmd0vs 30993	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ 0 ∈ (Base‘𝑊)) → ((0g‘𝐹) · 0 ) = 0 )
22	17, 21	syldan 595	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((0g‘𝐹) · 0 ) = 0 )
23	22	oveq2d 7164	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · ((0g‘𝐹) · 0 )) = (𝑋 · 0 ))
24	20, 23	eqtrd 2794	. 2 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → ((𝑋(.r‘𝐹)(0g‘𝐹)) · 0 ) = (𝑋 · 0 ))
25	8, 24, 22	3eqtr3d 2802	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ‘cfv 6333  (class class class)co 7148  Basecbs 16531  ._rcmulr 16614  Scalarcsca 16616   ·_𝑠 cvsca 16617  0_gc0g 16761  SRingcsrg 19313  SLModcslmd 30969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-iota 6292  df-fun 6335  df-fv 6341  df-riota 7106  df-ov 7151  df-0g 16763  df-mgm 17908  df-sgrp 17957  df-mnd 17968  df-cmn 18965  df-srg 19314  df-slmd 30970

This theorem is referenced by:  gsumvsca1  30995