Description: Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | elimph.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| elimph.5 | ⊢ 𝑍 = ( 0vec ‘ 𝑈 ) | ||
| elimph.6 | ⊢ 𝑈 ∈ CPreHilOLD | ||
| Assertion | elimph | ⊢ if ( 𝐴 ∈ 𝑋 , 𝐴 , 𝑍 ) ∈ 𝑋 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elimph.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| 2 | elimph.5 | ⊢ 𝑍 = ( 0vec ‘ 𝑈 ) | |
| 3 | elimph.6 | ⊢ 𝑈 ∈ CPreHilOLD | |
| 4 | 3 | phnvi | ⊢ 𝑈 ∈ NrmCVec |
| 5 | 1 2 4 | elimnv | ⊢ if ( 𝐴 ∈ 𝑋 , 𝐴 , 𝑍 ) ∈ 𝑋 |