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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 3514 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | thlval.k | . . 3 ⊢ 𝐾 = (toHL‘𝑊) | |
3 | fveq2 6672 | . . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊)) | |
4 | thlval.c | . . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
5 | 3, 4 | syl6eqr 2876 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶) |
6 | 5 | fveq2d 6676 | . . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶)) |
7 | thlval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐶) | |
8 | 6, 7 | syl6eqr 2876 | . . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼) |
9 | fveq2 6672 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊)) | |
10 | thlval.o | . . . . . . 7 ⊢ ⊥ = (ocv‘𝑊) | |
11 | 9, 10 | syl6eqr 2876 | . . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ ) |
12 | 11 | opeq2d 4812 | . . . . 5 ⊢ (ℎ = 𝑊 → 〈(oc‘ndx), (ocv‘ℎ)〉 = 〈(oc‘ndx), ⊥ 〉) |
13 | 8, 12 | oveq12d 7176 | . . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
14 | df-thl 20811 | . . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉)) | |
15 | ovex 7191 | . . . 4 ⊢ (𝐼 sSet 〈(oc‘ndx), ⊥ 〉) ∈ V | |
16 | 13, 14, 15 | fvmpt 6770 | . . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
17 | 2, 16 | syl5eq 2870 | . 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 2114 Vcvv 3496 〈cop 4575 ‘cfv 6357 (class class class)co 7158 ndxcnx 16482 sSet csts 16483 occoc 16575 toInccipo 17763 ocvcocv 20806 ClSubSpccss 20807 toHLcthl 20808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-thl 20811 |
This theorem is referenced by: thlbas 20842 thlle 20843 thloc 20845 |
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