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Theorem elimph 28247
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 = (0vec𝑈)
elimph.6 𝑈 ∈ CPreHilOLD
Assertion
Ref Expression
elimph if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋

Proof of Theorem elimph
StepHypRef Expression
1 elimph.1 . 2 𝑋 = (BaseSet‘𝑈)
2 elimph.5 . 2 𝑍 = (0vec𝑈)
3 elimph.6 . . 3 𝑈 ∈ CPreHilOLD
43phnvi 28243 . 2 𝑈 ∈ NrmCVec
51, 2, 4elimnv 28110 1 if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2107  ifcif 4307  cfv 6135  BaseSetcba 28013  0veccn0v 28015  CPreHilOLDccphlo 28239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-1st 7445  df-2nd 7446  df-grpo 27920  df-gid 27921  df-ablo 27972  df-vc 27986  df-nv 28019  df-va 28022  df-ba 28023  df-sm 28024  df-0v 28025  df-nmcv 28027  df-ph 28240
This theorem is referenced by:  ipdiri  28257  ipassi  28268  sii  28281
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