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Theorem nvsz 31240

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 2761	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 31217	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2761	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 31205	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvsz.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 31207	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		eqid 2761	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
8	7, 3	bafval 31206	. . . 4 ⊢ (BaseSet‘𝑈) = ran ( +_𝑣 ‘𝑈)
9		eqid 2761	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vcz 31177	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 592	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))) = (GId‘( +_𝑣 ‘𝑈)))
12		nvsz.6	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 31208	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 486	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	14	oveq2d 7436	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +_𝑣 ‘𝑈))))
16	11, 15, 14	3eqtr4d 2806	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 ℂcc 11198 GIdcgi 31092 CVec_OLDcvc 31160 NrmCVeccnv 31186 +_𝑣 cpv 31187 BaseSetcba 31188 ·_𝑠OLD cns 31189 0_veccn0v 31190

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-1st 8001 df-2nd 8002 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-ltxr 11348 df-grpo 31095 df-gid 31096 df-ginv 31097 df-ablo 31147 df-vc 31161 df-nv 31194 df-va 31197 df-ba 31198 df-sm 31199 df-0v 31200 df-nmcv 31202

This theorem is used by: nvmul0or 31252 nvnd 31290 dip0r 31319 0lno 31392

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