Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 28438 | . 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 1537 ∈ wcel 2114 ‘cfv 6355 ℝcr 10536 NrmCVeccnv 28361 BaseSetcba 28363 normCVcnmcv 28367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-1st 7689 df-2nd 7690 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-nmcv 28377 |
This theorem is referenced by: ip0i 28602 ip1ilem 28603 ipasslem10 28616 siilem1 28628 siii 28630 |
Copyright terms: Public domain | W3C validator |