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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 5715 | . . 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 18396 | . . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
9 | 4, 8 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
10 | 9 | simprld 769 | . . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵)) |
11 | 3, 10 | eleqtrrd 2835 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
12 | 9 | simprrd 771 | . . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵)) |
13 | 3, 12 | eleqtrrd 2835 | . 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 2105 〈cop 4634 × cxp 5674 dom cdm 5676 ‘cfv 6543 Basecbs 17151 Posetcpo 18270 joincjn 18274 meetcmee 18275 Latclat 18394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-dm 5686 df-iota 6495 df-fv 6551 df-lat 18395 |
This theorem is referenced by: latlej1 18411 latlej2 18412 latjle12 18413 latmle1 18427 latmle2 18428 latlem12 18429 |
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