nv0 - Metamath Proof Explorer

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Theorem nv0 31062

Description: Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nv0.1	⊢ 𝑋 = (BaseSet‘𝑈)
nv0.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
nv0.6	⊢ 𝑍 = (0_vec‘𝑈)

**Assertion**
Ref	Expression
nv0	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)

**Proof of Theorem nv0**
Step	Hyp	Ref	Expression
1		eqid 2765	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 31040	. . 3 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2765	. . . . 5 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 31028	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nv0.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 31030	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nv0.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 31029	. . . 4 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9		eqid 2765	. . . 4 ⊢ (GId‘( +_𝑣 ‘𝑈)) = (GId‘( +_𝑣 ‘𝑈))
10	4, 6, 8, 9	vc0 30999	. . 3 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +_𝑣 ‘𝑈)))
11	2, 10	sylan 592	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +_𝑣 ‘𝑈)))
12		nv0.6	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
13	3, 12	0vfval 31031	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
14	13	adantr 486	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘( +_𝑣 ‘𝑈)))
15	11, 14	eqtr4d 2803	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 0cc0 11117 GIdcgi 30915 CVec_OLDcvc 30983 NrmCVeccnv 31009 +_𝑣 cpv 31010 BaseSetcba 31011 ·_𝑠OLD cns 31012 0_veccn0v 31013

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-1st 7992 df-2nd 7993 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11262 df-mnf 11263 df-ltxr 11265 df-grpo 30918 df-gid 30919 df-ginv 30920 df-ablo 30970 df-vc 30984 df-nv 31017 df-va 31020 df-ba 31021 df-sm 31022 df-0v 31023 df-nmcv 31025

This theorem is used by: nvmul0or 31075 nvz0 31093 nvge0 31098 ipasslem1 31256 hlmul0 31334

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