MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvrel Structured version   Visualization version   GIF version

Theorem nvrel 28001
Description: The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nvrel Rel NrmCVec

Proof of Theorem nvrel
StepHypRef Expression
1 nvss 27992 . 2 NrmCVec ⊆ (CVecOLD × V)
2 relxp 5360 . 2 Rel (CVecOLD × V)
3 relss 5441 . 2 (NrmCVec ⊆ (CVecOLD × V) → (Rel (CVecOLD × V) → Rel NrmCVec))
41, 2, 3mp2 9 1 Rel NrmCVec
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3414  wss 3798   × cxp 5340  Rel wrel 5347  CVecOLDcvc 27957  NrmCVeccnv 27983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-opab 4936  df-xp 5348  df-rel 5349  df-oprab 6909  df-nv 27991
This theorem is referenced by:  nvop2  28007  nvop  28075  phrel  28214  bnrel  28267
  Copyright terms: Public domain W3C validator