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Theorem nvsz 30727

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 2739	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30704	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2739	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30692	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvsz.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30694	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		eqid 2739	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
8	7, 3	bafval 30693	. . . 4 ⊢ (BaseSet‘𝑈) = ran ( +_𝑣 ‘𝑈)
9		eqid 2739	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vcz 30664	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 586	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
12		nvsz.6	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 30695	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 481	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	14	oveq2d 7372	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))))
16	11, 15, 14	3eqtr4d 2784	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ‘cfv 6485 (class class class)co 7356 1^st c1st 7929 ℂcc 11027 GIdcgi 30579 CVec_OLDcvc 30647 NrmCVeccnv 30673 +_𝑣 cpv 30674 BaseSetcba 30675 ·_𝑠OLD cns 30676 0_veccn0v 30677

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-po 5526 df-so 5527 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-1st 7931 df-2nd 7932 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-ltxr 11175 df-grpo 30582 df-gid 30583 df-ginv 30584 df-ablo 30634 df-vc 30648 df-nv 30681 df-va 30684 df-ba 30685 df-sm 30686 df-0v 30687 df-nmcv 30689

This theorem is referenced by: nvmul0or 30739 nvnd 30777 dip0r 30806 0lno 30879

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