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Mirrors > Home > MPE Home > Th. List > nvscl | Structured version Visualization version GIF version |
Description: Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvscl.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvscl.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
nvscl | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 26854 | . 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2610 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 26842 | . . 3 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nvscl.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 26844 | . . 3 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nvscl.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 26843 | . . 3 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
9 | 4, 6, 8 | vccl 26802 | . 2 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
10 | 2, 9 | syl3an1 1351 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 ℂcc 9813 CVecOLDcvc 26797 NrmCVeccnv 26823 +𝑣 cpv 26824 BaseSetcba 26825 ·𝑠OLD cns 26826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-1st 7059 df-2nd 7060 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-nmcv 26839 |
This theorem is referenced by: nvmval2 26882 nvmf 26884 nvmdi 26887 nvnegneg 26888 nvpncan2 26892 nvaddsub4 26896 nvdif 26905 nvpi 26906 nvmtri 26910 nvabs 26911 nvge0 26912 imsmetlem 26929 smcnlem 26936 ipval2lem2 26943 4ipval2 26947 ipval3 26948 sspmval 26972 lnocoi 26996 lnomul 26999 0lno 27029 nmlno0lem 27032 nmblolbii 27038 blocnilem 27043 ip0i 27064 ip1ilem 27065 ipdirilem 27068 ipasslem1 27070 ipasslem2 27071 ipasslem4 27073 ipasslem5 27074 ipasslem8 27076 ipasslem9 27077 ipasslem10 27078 ipasslem11 27079 dipassr 27085 dipsubdir 27087 siilem1 27090 ipblnfi 27095 ubthlem2 27111 minvecolem2 27115 hhshsslem2 27509 |
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