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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 6906 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾)) | |
| 2 | islat.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
| 3 | 1, 2 | eqtr4di 2795 | . . . . 5 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = ∨ ) | 
| 4 | 3 | dmeqd 5916 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ ) | 
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
| 6 | islat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) | 
| 8 | 7 | sqxpeqd 5717 | . . . 4 ⊢ (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵)) | 
| 9 | 4, 8 | eqeq12d 2753 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∨ = (𝐵 × 𝐵))) | 
| 10 | fveq2 6906 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾)) | |
| 11 | islat.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
| 12 | 10, 11 | eqtr4di 2795 | . . . . 5 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = ∧ ) | 
| 13 | 12 | dmeqd 5916 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (meet‘𝑙) = dom ∧ ) | 
| 14 | 13, 8 | eqeq12d 2753 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∧ = (𝐵 × 𝐵))) | 
| 15 | 9, 14 | anbi12d 632 | . 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 18477 | . 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))} | |
| 17 | 15, 16 | elrab2 3695 | 1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 × cxp 5683 dom cdm 5685 ‘cfv 6561 Basecbs 17247 Posetcpo 18353 joincjn 18357 meetcmee 18358 Latclat 18476 | 
| 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-ext 2708 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 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-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-lat 18477 | 
| This theorem is referenced by: odulatb 18479 latcl2 18481 latlem 18482 latpos 18483 latjcom 18492 latmcom 18508 clatl 18553 toslat 48871 | 
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