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Theorem nvsz 28520

Description: Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvsz.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
nvsz.6	⊢ 𝑍 = (0_vec‘𝑈)

**Assertion**
Ref	Expression
nvsz	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

**Proof of Theorem nvsz**
Step	Hyp	Ref	Expression
1		eqid 2758	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 28497	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2758	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 28485	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvsz.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 28487	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		eqid 2758	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
8	7, 3	bafval 28486	. . . 4 ⊢ (BaseSet‘𝑈) = ran ( +_𝑣 ‘𝑈)
9		eqid 2758	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vcz 28457	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 583	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
12		nvsz.6	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 28488	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 484	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	14	oveq2d 7166	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))))
16	11, 15, 14	3eqtr4d 2803	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6335 (class class class)co 7150 1^st c1st 7691 ℂcc 10573 GIdcgi 28372 CVec_OLDcvc 28440 NrmCVeccnv 28466 +_𝑣 cpv 28467 BaseSetcba 28468 ·_𝑠OLD cns 28469 0_veccn0v 28470

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-op 4529 df-uni 4799 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-id 5430 df-po 5443 df-so 5444 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-1st 7693 df-2nd 7694 df-er 8299 df-en 8528 df-dom 8529 df-sdom 8530 df-pnf 10715 df-mnf 10716 df-ltxr 10718 df-grpo 28375 df-gid 28376 df-ginv 28377 df-ablo 28427 df-vc 28441 df-nv 28474 df-va 28477 df-ba 28478 df-sm 28479 df-0v 28480 df-nmcv 28482

This theorem is referenced by: nvmul0or 28532 nvnd 28570 dip0r 28599 0lno 28672

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