MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elimnv Structured version   Unicode version

Theorem elimnv 22167
Description: Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
elimnv.1  |-  X  =  ( BaseSet `  U )
elimnv.5  |-  Z  =  ( 0vec `  U
)
elimnv.9  |-  U  e.  NrmCVec
Assertion
Ref Expression
elimnv  |-  if ( A  e.  X ,  A ,  Z )  e.  X

Proof of Theorem elimnv
StepHypRef Expression
1 elimnv.9 . . 3  |-  U  e.  NrmCVec
2 elimnv.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 elimnv.5 . . . 4  |-  Z  =  ( 0vec `  U
)
42, 3nvzcl 22107 . . 3  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
51, 4ax-mp 8 . 2  |-  Z  e.  X
65elimel 3783 1  |-  if ( A  e.  X ,  A ,  Z )  e.  X
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   ifcif 3731   ` cfv 5446   NrmCVeccnv 22055   BaseSetcba 22057   0veccn0v 22059
This theorem is referenced by:  elimph  22313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21771  df-gid 21772  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
  Copyright terms: Public domain W3C validator