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Mirrors > Home > MPE Home > Th. List > thlval | Structured version Visualization version GIF version |
Description: Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.) |
Ref | Expression |
---|---|
thlval.k | ⊢ 𝐾 = (toHL‘𝑊) |
thlval.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
thlval.i | ⊢ 𝐼 = (toInc‘𝐶) |
thlval.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
thlval | ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3490 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | thlval.k | . . 3 ⊢ 𝐾 = (toHL‘𝑊) | |
3 | fveq2 6900 | . . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊)) | |
4 | thlval.c | . . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
5 | 3, 4 | eqtr4di 2785 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶) |
6 | 5 | fveq2d 6904 | . . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶)) |
7 | thlval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐶) | |
8 | 6, 7 | eqtr4di 2785 | . . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼) |
9 | fveq2 6900 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊)) | |
10 | thlval.o | . . . . . . 7 ⊢ ⊥ = (ocv‘𝑊) | |
11 | 9, 10 | eqtr4di 2785 | . . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ ) |
12 | 11 | opeq2d 4883 | . . . . 5 ⊢ (ℎ = 𝑊 → ⟨(oc‘ndx), (ocv‘ℎ)⟩ = ⟨(oc‘ndx), ⊥ ⟩) |
13 | 8, 12 | oveq12d 7442 | . . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩)) |
14 | df-thl 21602 | . . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩)) | |
15 | ovex 7457 | . . . 4 ⊢ (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩) ∈ V | |
16 | 13, 14, 15 | fvmpt 7008 | . . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩)) |
17 | 2, 16 | eqtrid 2779 | . 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩)) |
18 | 1, 17 | syl 17 | 1 ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ⟨cop 4636 ‘cfv 6551 (class class class)co 7424 sSet csts 17137 ndxcnx 17167 occoc 17246 toInccipo 18524 ocvcocv 21597 ClSubSpccss 21598 toHLcthl 21599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-thl 21602 |
This theorem is referenced by: thlbas 21633 thlbasOLD 21634 thlle 21635 thlleOLD 21636 thloc 21638 |
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