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Theorem nvsid 30145

Description: Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvscl.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvscl.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
nvsid	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

**Proof of Theorem nvsid**
Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30133	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2730	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30121	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvscl.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30123	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvscl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30122	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vcidOLD 30082	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
10	2, 9	sylan 578	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ‘cfv 6544 (class class class)co 7413 1^st c1st 7977 1c1 11115 CVec_OLDcvc 30076 NrmCVeccnv 30102 +_𝑣 cpv 30103 BaseSetcba 30104 ·_𝑠OLD cns 30105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-oprab 7417 df-1st 7979 df-2nd 7980 df-vc 30077 df-nv 30110 df-va 30113 df-ba 30114 df-sm 30115 df-0v 30116 df-nmcv 30118

This theorem is referenced by: nvmul0or 30168 nvpi 30185 nvge0 30191 ipval2lem3 30223 ipval2 30225 ipidsq 30228 lnoadd 30276 ip1ilem 30344 ip2i 30346 ipdirilem 30347 ipasslem1 30349 ipasslem4 30352 ipasslem10 30357 minvecolem2 30393 hlmulid 30423

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