nvscl - Metamath Proof Explorer

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Theorem nvscl 30528

Description: Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
nvscl	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

**Proof of Theorem nvscl**
Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30517	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2729	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30505	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvscl.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30507	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvscl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30506	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vccl 30465	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
10	2, 9	syl3an1 1163	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 ℂcc 11042 CVec_OLDcvc 30460 NrmCVeccnv 30486 +_𝑣 cpv 30487 BaseSetcba 30488 ·_𝑠OLD cns 30489

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-1st 7947 df-2nd 7948 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-nmcv 30502

This theorem is referenced by: nvmval2 30545 nvmf 30547 nvmdi 30550 nvnegneg 30551 nvpncan2 30555 nvaddsub4 30559 nvdif 30568 nvpi 30569 nvmtri 30573 nvabs 30574 nvge0 30575 imsmetlem 30592 smcnlem 30599 ipval2lem2 30606 4ipval2 30610 ipval3 30611 sspmval 30635 lnocoi 30659 lnomul 30662 0lno 30692 nmlno0lem 30695 nmblolbii 30701 blocnilem 30706 ip0i 30727 ip1ilem 30728 ipdirilem 30731 ipasslem1 30733 ipasslem2 30734 ipasslem4 30736 ipasslem5 30737 ipasslem8 30739 ipasslem9 30740 ipasslem10 30741 ipasslem11 30742 dipassr 30748 dipsubdir 30750 siilem1 30753 ipblnfi 30757 ubthlem2 30773 minvecolem2 30777 hhshsslem2 31170

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