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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 = (CSubSp‘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 3185 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | thlval.k | . . 3 ⊢ 𝐾 = (toHL‘𝑊) | |
3 | fveq2 6103 | . . . . . . . 8 ⊢ (ℎ = 𝑊 → (CSubSp‘ℎ) = (CSubSp‘𝑊)) | |
4 | thlval.c | . . . . . . . 8 ⊢ 𝐶 = (CSubSp‘𝑊) | |
5 | 3, 4 | syl6eqr 2662 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (CSubSp‘ℎ) = 𝐶) |
6 | 5 | fveq2d 6107 | . . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(CSubSp‘ℎ)) = (toInc‘𝐶)) |
7 | thlval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐶) | |
8 | 6, 7 | syl6eqr 2662 | . . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(CSubSp‘ℎ)) = 𝐼) |
9 | fveq2 6103 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊)) | |
10 | thlval.o | . . . . . . 7 ⊢ ⊥ = (ocv‘𝑊) | |
11 | 9, 10 | syl6eqr 2662 | . . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ ) |
12 | 11 | opeq2d 4347 | . . . . 5 ⊢ (ℎ = 𝑊 → 〈(oc‘ndx), (ocv‘ℎ)〉 = 〈(oc‘ndx), ⊥ 〉) |
13 | 8, 12 | oveq12d 6567 | . . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(CSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
14 | df-thl 19828 | . . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(CSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉)) | |
15 | ovex 6577 | . . . 4 ⊢ (𝐼 sSet 〈(oc‘ndx), ⊥ 〉) ∈ V | |
16 | 13, 14, 15 | fvmpt 6191 | . . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
17 | 2, 16 | syl5eq 2656 | . 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 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 sSet csts 15693 occoc 15776 toInccipo 16974 ocvcocv 19823 CSubSpccss 19824 toHLcthl 19825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-thl 19828 |
This theorem is referenced by: thlbas 19859 thlle 19860 thloc 19862 |
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