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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 3501 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
| 2 | thlval.k | . . 3 ⊢ 𝐾 = (toHL‘𝑊) | |
| 3 | fveq2 6906 | . . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊)) | |
| 4 | thlval.c | . . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶) |
| 6 | 5 | fveq2d 6910 | . . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶)) |
| 7 | thlval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐶) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼) |
| 9 | fveq2 6906 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊)) | |
| 10 | thlval.o | . . . . . . 7 ⊢ ⊥ = (ocv‘𝑊) | |
| 11 | 9, 10 | eqtr4di 2795 | . . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ ) |
| 12 | 11 | opeq2d 4880 | . . . . 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 21683 | . . . 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 2789 | . 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 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 occoc 17305 toInccipo 18572 ocvcocv 21678 ClSubSpccss 21679 toHLcthl 21680 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-thl 21683 |
| This theorem is referenced by: thlbas 21714 thlbasOLD 21715 thlle 21716 thlleOLD 21717 thloc 21719 |
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