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Mirrors > Home > MPE Home > Th. List > islat | Structured version Visualization version GIF version |
Description: The predicate "is a lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
islat.b | ⊢ 𝐵 = (Base‘𝐾) |
islat.j | ⊢ ∨ = (join‘𝐾) |
islat.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
islat | ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6774 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾)) | |
2 | islat.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
3 | 1, 2 | eqtr4di 2796 | . . . . 5 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = ∨ ) |
4 | 3 | dmeqd 5814 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ ) |
5 | fveq2 6774 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
6 | islat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 5, 6 | eqtr4di 2796 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) |
8 | 7 | sqxpeqd 5621 | . . . 4 ⊢ (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵)) |
9 | 4, 8 | eqeq12d 2754 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∨ = (𝐵 × 𝐵))) |
10 | fveq2 6774 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾)) | |
11 | islat.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
12 | 10, 11 | eqtr4di 2796 | . . . . 5 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = ∧ ) |
13 | 12 | dmeqd 5814 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (meet‘𝑙) = dom ∧ ) |
14 | 13, 8 | eqeq12d 2754 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∧ = (𝐵 × 𝐵))) |
15 | 9, 14 | anbi12d 631 | . 2 ⊢ (𝑙 = 𝐾 → ((dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙))) ↔ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
16 | df-lat 18150 | . 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))} | |
17 | 15, 16 | elrab2 3627 | 1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 × cxp 5587 dom cdm 5589 ‘cfv 6433 Basecbs 16912 Posetcpo 18025 joincjn 18029 meetcmee 18030 Latclat 18149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-dm 5599 df-iota 6391 df-fv 6441 df-lat 18150 |
This theorem is referenced by: odulatb 18152 latcl2 18154 latlem 18155 latpos 18156 latjcom 18165 latmcom 18181 clatl 18226 toslat 46268 |
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