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Theorem elimph 29811
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 = (BaseSetβ€˜π‘ˆ)
elimph.5 𝑍 = (0vecβ€˜π‘ˆ)
elimph.6 π‘ˆ ∈ CPreHilOLD
Assertion
Ref Expression
elimph if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋

Proof of Theorem elimph
StepHypRef Expression
1 elimph.1 . 2 𝑋 = (BaseSetβ€˜π‘ˆ)
2 elimph.5 . 2 𝑍 = (0vecβ€˜π‘ˆ)
3 elimph.6 . . 3 π‘ˆ ∈ CPreHilOLD
43phnvi 29807 . 2 π‘ˆ ∈ NrmCVec
51, 2, 4elimnv 29674 1 if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  ifcif 4490  β€˜cfv 6500  BaseSetcba 29577  0veccn0v 29579  CPreHilOLDccphlo 29803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-1st 7925  df-2nd 7926  df-grpo 29484  df-gid 29485  df-ablo 29536  df-vc 29550  df-nv 29583  df-va 29586  df-ba 29587  df-sm 29588  df-0v 29589  df-nmcv 29591  df-ph 29804
This theorem is referenced by:  ipdiri  29821  ipassi  29832  sii  29845
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