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Theorem elimnv 29863
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 = (0vec𝑈)
elimnv.9 𝑈 ∈ NrmCVec
Assertion
Ref Expression
elimnv if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋

Proof of Theorem elimnv
StepHypRef Expression
1 elimnv.9 . . 3 𝑈 ∈ NrmCVec
2 elimnv.1 . . . 4 𝑋 = (BaseSet‘𝑈)
3 elimnv.5 . . . 4 𝑍 = (0vec𝑈)
42, 3nvzcl 29814 . . 3 (𝑈 ∈ NrmCVec → 𝑍𝑋)
51, 4ax-mp 5 . 2 𝑍𝑋
65elimel 4592 1 if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  ifcif 4523  cfv 6533  NrmCVeccnv 29764  BaseSetcba 29766  0veccn0v 29768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-1st 7959  df-2nd 7960  df-grpo 29673  df-gid 29674  df-ablo 29725  df-vc 29739  df-nv 29772  df-va 29775  df-ba 29776  df-sm 29777  df-0v 29778  df-nmcv 29780
This theorem is referenced by:  elimph  30000
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