Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | opelxpi 5072 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
4 | 1, 2, 3 | syl2anc 691 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
5 | latcl2.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) | |
6 | latcl2.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
7 | latcl2.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
8 | latcl2.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
9 | 6, 7, 8 | islat 16870 | . . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
10 | 5, 9 | sylib 207 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
11 | simprl 790 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))) → dom ∨ = (𝐵 × 𝐵)) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵)) |
13 | 4, 12 | eleqtrrd 2691 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
14 | 10 | simprrd 793 | . . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵)) |
15 | 4, 14 | eleqtrrd 2691 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
16 | 13, 15 | jca 553 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 dom cdm 5038 ‘cfv 5804 Basecbs 15695 Posetcpo 16763 joincjn 16767 meetcmee 16768 Latclat 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-lat 16869 |
This theorem is referenced by: latlej1 16883 latlej2 16884 latjle12 16885 latmle1 16899 latmle2 16900 latlem12 16901 |
Copyright terms: Public domain | W3C validator |