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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 5589 | . . 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 17939 | . . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
9 | 4, 8 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
10 | 9 | simprld 772 | . . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵)) |
11 | 3, 10 | eleqtrrd 2841 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
12 | 9 | simprrd 774 | . . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵)) |
13 | 3, 12 | eleqtrrd 2841 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
14 | 11, 13 | jca 515 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 〈cop 4547 × cxp 5549 dom cdm 5551 ‘cfv 6380 Basecbs 16760 Posetcpo 17814 joincjn 17818 meetcmee 17819 Latclat 17937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-xp 5557 df-dm 5561 df-iota 6338 df-fv 6388 df-lat 17938 |
This theorem is referenced by: latlej1 17954 latlej2 17955 latjle12 17956 latmle1 17970 latmle2 17971 latlem12 17972 |
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