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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 5595 | . . 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 17659 | . . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
9 | 4, 8 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
10 | 9 | simprld 770 | . . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵)) |
11 | 3, 10 | eleqtrrd 2918 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
12 | 9 | simprrd 772 | . . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵)) |
13 | 3, 12 | eleqtrrd 2918 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
14 | 11, 13 | jca 514 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 dom cdm 5557 ‘cfv 6357 Basecbs 16485 Posetcpo 17552 joincjn 17556 meetcmee 17557 Latclat 17657 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-dm 5567 df-iota 6316 df-fv 6365 df-lat 17658 |
This theorem is referenced by: latlej1 17672 latlej2 17673 latjle12 17674 latmle1 17688 latmle2 17689 latlem12 17690 |
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