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Mirrors > Home > MPE Home > Th. List > phnvi | Structured version Visualization version GIF version |
Description: Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
phnvi.1 | ⊢ 𝑈 ∈ CPreHilOLD |
Ref | Expression |
---|---|
phnvi | ⊢ 𝑈 ∈ NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phnvi.1 | . 2 ⊢ 𝑈 ∈ CPreHilOLD | |
2 | phnv 27053 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 NrmCVeccnv 26823 CPreHilOLDccphlo 27051 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-ph 27052 |
This theorem is referenced by: elimph 27059 ip0i 27064 ip1ilem 27065 ip2i 27067 ipdirilem 27068 ipasslem1 27070 ipasslem2 27071 ipasslem4 27073 ipasslem5 27074 ipasslem7 27075 ipasslem8 27076 ipasslem9 27077 ipasslem10 27078 ipasslem11 27079 ip2dii 27083 pythi 27089 siilem1 27090 siilem2 27091 siii 27092 ipblnfi 27095 ip2eqi 27096 ajfuni 27099 |
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