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Theorem elimph 21453
Description: Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
elimph.1  |-  X  =  ( BaseSet `  U )
elimph.5  |-  Z  =  ( 0vec `  U
)
elimph.6  |-  U  e.  CPreHil
OLD
Assertion
Ref Expression
elimph  |-  if ( A  e.  X ,  A ,  Z )  e.  X

Proof of Theorem elimph
StepHypRef Expression
1 elimph.1 . 2  |-  X  =  ( BaseSet `  U )
2 elimph.5 . 2  |-  Z  =  ( 0vec `  U
)
3 elimph.6 . . 3  |-  U  e.  CPreHil
OLD
43phnvi 21449 . 2  |-  U  e.  NrmCVec
51, 2, 4elimnv 21307 1  |-  if ( A  e.  X ,  A ,  Z )  e.  X
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701   ifcif 3599   ` cfv 5292   BaseSetcba 21197   0veccn0v 21199   CPreHil OLDccphlo 21445
This theorem is referenced by:  ipdiri  21463  ipassi  21474  sii  21487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-1st 6164  df-2nd 6165  df-riota 6346  df-grpo 20911  df-gid 20912  df-ablo 21002  df-vc 21157  df-nv 21203  df-va 21206  df-ba 21207  df-sm 21208  df-0v 21209  df-nmcv 21211  df-ph 21446
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