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| Description: The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) | 
| Ref | Expression | 
|---|---|
| ishil.k | ⊢ 𝐾 = (proj‘𝐻) | 
| ishil.c | ⊢ 𝐶 = (ClSubSp‘𝐻) | 
| Ref | Expression | 
|---|---|
| ishil | ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6906 | . . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻)) | |
| 2 | ishil.k | . . . . 5 ⊢ 𝐾 = (proj‘𝐻) | |
| 3 | 1, 2 | eqtr4di 2795 | . . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾) | 
| 4 | 3 | dmeqd 5916 | . . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾) | 
| 5 | fveq2 6906 | . . . 4 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = (ClSubSp‘𝐻)) | |
| 6 | ishil.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝐻) | |
| 7 | 5, 6 | eqtr4di 2795 | . . 3 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = 𝐶) | 
| 8 | 4, 7 | eqeq12d 2753 | . 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (ClSubSp‘ℎ) ↔ dom 𝐾 = 𝐶)) | 
| 9 | df-hil 21724 | . 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)} | |
| 10 | 8, 9 | elrab2 3695 | 1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 dom cdm 5685 ‘cfv 6561 PreHilcphl 21642 ClSubSpccss 21679 projcpj 21720 Hilchil 21721 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-hil 21724 | 
| This theorem is referenced by: ishil2 21739 hlhil 25477 | 
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