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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 3499 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | thlval.k | . . 3 ⊢ 𝐾 = (toHL‘𝑊) | |
3 | fveq2 6907 | . . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊)) | |
4 | thlval.c | . . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
5 | 3, 4 | eqtr4di 2793 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶) |
6 | 5 | fveq2d 6911 | . . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶)) |
7 | thlval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐶) | |
8 | 6, 7 | eqtr4di 2793 | . . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼) |
9 | fveq2 6907 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊)) | |
10 | thlval.o | . . . . . . 7 ⊢ ⊥ = (ocv‘𝑊) | |
11 | 9, 10 | eqtr4di 2793 | . . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ ) |
12 | 11 | opeq2d 4885 | . . . . 5 ⊢ (ℎ = 𝑊 → 〈(oc‘ndx), (ocv‘ℎ)〉 = 〈(oc‘ndx), ⊥ 〉) |
13 | 8, 12 | oveq12d 7449 | . . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
14 | df-thl 21701 | . . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉)) | |
15 | ovex 7464 | . . . 4 ⊢ (𝐼 sSet 〈(oc‘ndx), ⊥ 〉) ∈ V | |
16 | 13, 14, 15 | fvmpt 7016 | . . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
17 | 2, 16 | eqtrid 2787 | . 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 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 ndxcnx 17227 occoc 17306 toInccipo 18585 ocvcocv 21696 ClSubSpccss 21697 toHLcthl 21698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-thl 21701 |
This theorem is referenced by: thlbas 21732 thlbasOLD 21733 thlle 21734 thlleOLD 21735 thloc 21737 |
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