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Mirrors > Home > MPE Home > Th. List > nvcli | Structured version Visualization version GIF version |
Description: The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvf.6 | ⊢ 𝑁 = (normCV‘𝑈) |
nvcli.9 | ⊢ 𝑈 ∈ NrmCVec |
nvcli.7 | ⊢ 𝐴 ∈ 𝑋 |
Ref | Expression |
---|---|
nvcli | ⊢ (𝑁‘𝐴) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvcli.9 | . 2 ⊢ 𝑈 ∈ NrmCVec | |
2 | nvcli.7 | . 2 ⊢ 𝐴 ∈ 𝑋 | |
3 | nvf.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | nvf.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 3, 4 | nvcl 28436 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
6 | 1, 2, 5 | mp2an 690 | 1 ⊢ (𝑁‘𝐴) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ‘cfv 6348 ℝcr 10529 NrmCVeccnv 28359 BaseSetcba 28361 normCVcnmcv 28365 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-1st 7682 df-2nd 7683 df-vc 28334 df-nv 28367 df-va 28370 df-ba 28371 df-sm 28372 df-0v 28373 df-nmcv 28375 |
This theorem is referenced by: ip0i 28600 ip1ilem 28601 ipasslem10 28614 siilem1 28626 siii 28628 |
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