nv0 - Metamath Proof Explorer

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Theorem nv0 28585

Description: Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nv0.1	⊢ 𝑋 = (BaseSet‘𝑈)
nv0.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
nv0.6	⊢ 𝑍 = (0_vec‘𝑈)

**Assertion**
Ref	Expression
nv0	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)

**Proof of Theorem nv0**
Step	Hyp	Ref	Expression
1		eqid 2739	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 28563	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2739	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 28551	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nv0.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 28553	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nv0.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 28552	. . . 4 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9		eqid 2739	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vc0 28522	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 583	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +_𝑣 ‘𝑈)))
12		nv0.6	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 28554	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 484	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	11, 14	eqtr4d 2777	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6350 (class class class)co 7183 1^st c1st 7725 0cc0 10628 GIdcgi 28438 CVec_OLDcvc 28506 NrmCVeccnv 28532 +_𝑣 cpv 28533 BaseSetcba 28534 ·_𝑠OLD cns 28535 0_veccn0v 28536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-1st 7727 df-2nd 7728 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-ltxr 10771 df-grpo 28441 df-gid 28442 df-ginv 28443 df-ablo 28493 df-vc 28507 df-nv 28540 df-va 28543 df-ba 28544 df-sm 28545 df-0v 28546 df-nmcv 28548

This theorem is referenced by: nvmul0or 28598 nvz0 28616 nvge0 28621 ipasslem1 28779 hlmul0 28857

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