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| Mirrors > Home > MPE Home > Th. List > latcl2 | Structured version Visualization version GIF version | ||
| Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.) |
| Ref | Expression |
|---|---|
| latcl2.b | ⊢ 𝐵 = (Base‘𝐾) |
| latcl2.j | ⊢ ∨ = (join‘𝐾) |
| latcl2.m | ⊢ ∧ = (meet‘𝐾) |
| latcl2.k | ⊢ (𝜑 → 𝐾 ∈ Lat) |
| latcl2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| latcl2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| latcl2 | ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latcl2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 2 | latcl2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 3 | 1, 2 | opelxpd 5724 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
| 4 | latcl2.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) | |
| 5 | latcl2.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 6 | latcl2.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
| 7 | latcl2.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
| 8 | 5, 6, 7 | islat 18478 | . . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
| 9 | 4, 8 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
| 10 | 9 | simprld 772 | . . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵)) |
| 11 | 3, 10 | eleqtrrd 2844 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
| 12 | 9 | simprrd 774 | . . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵)) |
| 13 | 3, 12 | eleqtrrd 2844 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
| 14 | 11, 13 | jca 511 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 × 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 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 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-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-lat 18477 |
| This theorem is referenced by: latlej1 18493 latlej2 18494 latjle12 18495 latmle1 18509 latmle2 18510 latlem12 18511 |
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