nv2 - Metamath Proof Explorer

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Theorem nv2 30965

Description: A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvdi.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvdi.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
nvdi.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
nv2	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

**Proof of Theorem nv2**
Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30948	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		nvdi.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4	3	vafval 30936	. . 3 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
5		nvdi.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30938	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvdi.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30937	. . 3 ⊢ 𝑋 = ran 𝐺
9	4, 6, 8	vc2OLD 30901	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
10	2, 9	sylan 591	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 2c2 12296 CVec_OLDcvc 30891 NrmCVeccnv 30917 +_𝑣 cpv 30918 BaseSetcba 30919 ·_𝑠OLD cns 30920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pr 5406 ax-un 7734 ax-1cn 11159

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 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 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-oprab 7416 df-1st 7987 df-2nd 7988 df-2 12304 df-vc 30892 df-nv 30925 df-va 30928 df-ba 30929 df-sm 30930 df-0v 30931 df-nmcv 30933

This theorem is referenced by: ipidsq 31043 minvecolem2 31208

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