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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 6674 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾)) | |
2 | islat.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
3 | 1, 2 | eqtr4di 2791 | . . . . 5 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = ∨ ) |
4 | 3 | dmeqd 5748 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ ) |
5 | fveq2 6674 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
6 | islat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 5, 6 | eqtr4di 2791 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) |
8 | 7 | sqxpeqd 5557 | . . . 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 6674 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾)) | |
11 | islat.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
12 | 10, 11 | eqtr4di 2791 | . . . . 5 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = ∧ ) |
13 | 12 | dmeqd 5748 | . . . 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 634 | . 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 17772 | . 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))} | |
17 | 15, 16 | elrab2 3591 | 1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 × cxp 5523 dom cdm 5525 ‘cfv 6339 Basecbs 16586 Posetcpo 17666 joincjn 17670 meetcmee 17671 Latclat 17771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-xp 5531 df-dm 5535 df-iota 6297 df-fv 6347 df-lat 17772 |
This theorem is referenced by: latcl2 17774 latlem 17775 latpos 17776 latjcom 17785 latmcom 17801 clatl 17842 odulatb 17869 |
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