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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 3212 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | thlval.k | . . 3 ⊢ 𝐾 = (toHL‘𝑊) | |
3 | fveq2 6191 | . . . . . . . 8 ⊢ (ℎ = 𝑊 → (CSubSp‘ℎ) = (CSubSp‘𝑊)) | |
4 | thlval.c | . . . . . . . 8 ⊢ 𝐶 = (CSubSp‘𝑊) | |
5 | 3, 4 | syl6eqr 2674 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (CSubSp‘ℎ) = 𝐶) |
6 | 5 | fveq2d 6195 | . . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(CSubSp‘ℎ)) = (toInc‘𝐶)) |
7 | thlval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐶) | |
8 | 6, 7 | syl6eqr 2674 | . . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(CSubSp‘ℎ)) = 𝐼) |
9 | fveq2 6191 | . . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊)) | |
10 | thlval.o | . . . . . . 7 ⊢ ⊥ = (ocv‘𝑊) | |
11 | 9, 10 | syl6eqr 2674 | . . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ ) |
12 | 11 | opeq2d 4409 | . . . . 5 ⊢ (ℎ = 𝑊 → 〈(oc‘ndx), (ocv‘ℎ)〉 = 〈(oc‘ndx), ⊥ 〉) |
13 | 8, 12 | oveq12d 6668 | . . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(CSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
14 | df-thl 20009 | . . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(CSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉)) | |
15 | ovex 6678 | . . . 4 ⊢ (𝐼 sSet 〈(oc‘ndx), ⊥ 〉) ∈ V | |
16 | 13, 14, 15 | fvmpt 6282 | . . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) |
17 | 2, 16 | syl5eq 2668 | . 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 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 sSet csts 15855 occoc 15949 toInccipo 17151 ocvcocv 20004 CSubSpccss 20005 toHLcthl 20006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-thl 20009 |
This theorem is referenced by: thlbas 20040 thlle 20041 thloc 20043 |
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