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Description: Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex ). Note that ~H is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as Theorem hhba . (Contributed by NM, 31-May-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-hba | ⊢ ℋ = ( BaseSet ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | chba | ⊢ ℋ | |
| 1 | cba | ⊢ BaseSet | |
| 2 | cva | ⊢ +ℎ | |
| 3 | csm | ⊢ ·ℎ | |
| 4 | 2 3 | cop | ⊢ 〈 +ℎ , ·ℎ 〉 |
| 5 | cno | ⊢ normℎ | |
| 6 | 4 5 | cop | ⊢ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 |
| 7 | 6 1 | cfv | ⊢ ( BaseSet ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |
| 8 | 0 7 | wceq | ⊢ ℋ = ( BaseSet ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |