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Theorem nvsz 28049

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 2826	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 28026	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2826	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 28014	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvsz.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 28016	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		eqid 2826	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
8	7, 3	bafval 28015	. . . 4 ⊢ (BaseSet‘𝑈) = ran ( +_𝑣 ‘𝑈)
9		eqid 2826	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vcz 27986	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 577	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
12		nvsz.6	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 28017	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 474	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	14	oveq2d 6922	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))))
16	11, 15, 14	3eqtr4d 2872	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ‘cfv 6124 (class class class)co 6906 1^st c1st 7427 ℂcc 10251 GIdcgi 27901 CVec_OLDcvc 27969 NrmCVeccnv 27995 +_𝑣 cpv 27996 BaseSetcba 27997 ·_𝑠OLD cns 27998 0_veccn0v 27999

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-1st 7429 df-2nd 7430 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-ltxr 10397 df-grpo 27904 df-gid 27905 df-ginv 27906 df-ablo 27956 df-vc 27970 df-nv 28003 df-va 28006 df-ba 28007 df-sm 28008 df-0v 28009 df-nmcv 28011

This theorem is referenced by: nvmul0or 28061 nvnd 28099 dip0r 28128 0lno 28201

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