Theorem elimph 28591

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 ∈ CPreHilOLD

Assertion

Ref	Expression
elimph	⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋

Proof of Theorem elimph

Step	Hyp	Ref	Expression
1		elimph.1	. 2 ⊢ 𝑋 = (BaseSet‘𝑈)
2		elimph.5	. 2 ⊢ 𝑍 = (0vec‘𝑈)
3		elimph.6	. . 3 ⊢ 𝑈 ∈ CPreHilOLD
4	3	phnvi 28587	. 2 ⊢ 𝑈 ∈ NrmCVec
5	1, 2, 4	elimnv 28454	1 ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋

Syntax hints:   = wceq 1533   ∈ wcel 2110  ifcif 4466  ‘cfv 6349  BaseSetcba 28357  0_veccn0v 28359  CPreHil_OLDccphlo 28583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-1st 7683  df-2nd 7684  df-grpo 28264  df-gid 28265  df-ablo 28316  df-vc 28330  df-nv 28363  df-va 28366  df-ba 28367  df-sm 28368  df-0v 28369  df-nmcv 28371  df-ph 28584

This theorem is referenced by:  ipdiri  28601  ipassi  28612  sii  28625