nv0 - Metamath Proof Explorer

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Theorem nv0 30573

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 2730	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30551	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2730	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30539	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nv0.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30541	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nv0.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30540	. . . 4 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9		eqid 2730	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vc0 30510	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 580	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +_𝑣 ‘𝑈)))
12		nv0.6	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 30542	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 480	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	11, 14	eqtr4d 2768	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 0cc0 11075 GIdcgi 30426 CVec_OLDcvc 30494 NrmCVeccnv 30520 +_𝑣 cpv 30521 BaseSetcba 30522 ·_𝑠OLD cns 30523 0_veccn0v 30524

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-1st 7971 df-2nd 7972 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-ltxr 11220 df-grpo 30429 df-gid 30430 df-ginv 30431 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-nmcv 30536

This theorem is referenced by: nvmul0or 30586 nvz0 30604 nvge0 30609 ipasslem1 30767 hlmul0 30845

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