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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 3487 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | thlval.k | . . 3 ⊢ 𝐾 = (toHL‘𝑊) | |
3 | fveq2 6884 | . . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊)) | |
4 | thlval.c | . . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
5 | 3, 4 | eqtr4di 2784 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶) |
6 | 5 | fveq2d 6888 | . . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶)) |
7 | thlval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐶) | |
8 | 6, 7 | eqtr4di 2784 | . . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼) |
9 | fveq2 6884 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊)) | |
10 | thlval.o | . . . . . . 7 ⊢ ⊥ = (ocv‘𝑊) | |
11 | 9, 10 | eqtr4di 2784 | . . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ ) |
12 | 11 | opeq2d 4875 | . . . . 5 ⊢ (ℎ = 𝑊 → ⟨(oc‘ndx), (ocv‘ℎ)⟩ = ⟨(oc‘ndx), ⊥ ⟩) |
13 | 8, 12 | oveq12d 7422 | . . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩)) |
14 | df-thl 21554 | . . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩)) | |
15 | ovex 7437 | . . . 4 ⊢ (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩) ∈ V | |
16 | 13, 14, 15 | fvmpt 6991 | . . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩)) |
17 | 2, 16 | eqtrid 2778 | . 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 3468 ⟨cop 4629 ‘cfv 6536 (class class class)co 7404 sSet csts 17103 ndxcnx 17133 occoc 17212 toInccipo 18490 ocvcocv 21549 ClSubSpccss 21550 toHLcthl 21551 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-thl 21554 |
This theorem is referenced by: thlbas 21585 thlbasOLD 21586 thlle 21587 thlleOLD 21588 thloc 21590 |
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