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Mirrors > Home > MPE Home > Th. List > ishil | Structured version Visualization version GIF version |
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 6645 | . . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻)) | |
2 | ishil.k | . . . . 5 ⊢ 𝐾 = (proj‘𝐻) | |
3 | 1, 2 | eqtr4di 2851 | . . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾) |
4 | 3 | dmeqd 5738 | . . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾) |
5 | fveq2 6645 | . . . 4 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = (ClSubSp‘𝐻)) | |
6 | ishil.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝐻) | |
7 | 5, 6 | eqtr4di 2851 | . . 3 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = 𝐶) |
8 | 4, 7 | eqeq12d 2814 | . 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (ClSubSp‘ℎ) ↔ dom 𝐾 = 𝐶)) |
9 | df-hil 20393 | . 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)} | |
10 | 8, 9 | elrab2 3631 | 1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 dom cdm 5519 ‘cfv 6324 PreHilcphl 20313 ClSubSpccss 20350 projcpj 20389 Hilchil 20390 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 df-hil 20393 |
This theorem is referenced by: ishil2 20408 hlhil 24047 |
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