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Theorem nvsz 30683

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 2737	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30660	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2737	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30648	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvsz.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30650	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		eqid 2737	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
8	7, 3	bafval 30649	. . . 4 ⊢ (BaseSet‘𝑈) = ran ( +_𝑣 ‘𝑈)
9		eqid 2737	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vcz 30620	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 580	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
12		nvsz.6	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 30651	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 480	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	14	oveq2d 7454	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))))
16	11, 15, 14	3eqtr4d 2787	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ‘cfv 6569 (class class class)co 7438 1^st c1st 8020 ℂcc 11160 GIdcgi 30535 CVec_OLDcvc 30603 NrmCVeccnv 30629 +_𝑣 cpv 30630 BaseSetcba 30631 ·_𝑠OLD cns 30632 0_veccn0v 30633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-id 5587 df-po 5601 df-so 5602 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-1st 8022 df-2nd 8023 df-er 8753 df-en 8994 df-dom 8995 df-sdom 8996 df-pnf 11304 df-mnf 11305 df-ltxr 11307 df-grpo 30538 df-gid 30539 df-ginv 30540 df-ablo 30590 df-vc 30604 df-nv 30637 df-va 30640 df-ba 30641 df-sm 30642 df-0v 30643 df-nmcv 30645

This theorem is referenced by: nvmul0or 30695 nvnd 30733 dip0r 30762 0lno 30835

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