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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 5618 | . . 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 18066 | . . . . 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 768 | . . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵)) |
11 | 3, 10 | eleqtrrd 2842 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
12 | 9 | simprrd 770 | . . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵)) |
13 | 3, 12 | eleqtrrd 2842 | . 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 1539 ∈ wcel 2108 〈cop 4564 × cxp 5578 dom cdm 5580 ‘cfv 6418 Basecbs 16840 Posetcpo 17940 joincjn 17944 meetcmee 17945 Latclat 18064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-xp 5586 df-dm 5590 df-iota 6376 df-fv 6426 df-lat 18065 |
This theorem is referenced by: latlej1 18081 latlej2 18082 latjle12 18083 latmle1 18097 latmle2 18098 latlem12 18099 |
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