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Theorem latcl2 16988
Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
latcl2.b 𝐵 = (Base‘𝐾)
latcl2.j = (join‘𝐾)
latcl2.m = (meet‘𝐾)
latcl2.k (𝜑𝐾 ∈ Lat)
latcl2.x (𝜑𝑋𝐵)
latcl2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
latcl2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑋, 𝑌⟩ ∈ dom ))

Proof of Theorem latcl2
StepHypRef Expression
1 latcl2.x . . . 4 (𝜑𝑋𝐵)
2 latcl2.y . . . 4 (𝜑𝑌𝐵)
3 opelxpi 5118 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
41, 2, 3syl2anc 692 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
5 latcl2.k . . . . 5 (𝜑𝐾 ∈ Lat)
6 latcl2.b . . . . . 6 𝐵 = (Base‘𝐾)
7 latcl2.j . . . . . 6 = (join‘𝐾)
8 latcl2.m . . . . . 6 = (meet‘𝐾)
96, 7, 8islat 16987 . . . . 5 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
105, 9sylib 208 . . . 4 (𝜑 → (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
11 simprl 793 . . . 4 ((𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))) → dom = (𝐵 × 𝐵))
1210, 11syl 17 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
134, 12eleqtrrd 2701 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
1410simprrd 796 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
154, 14eleqtrrd 2701 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
1613, 15jca 554 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑋, 𝑌⟩ ∈ dom ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cop 4161   × cxp 5082  dom cdm 5084  cfv 5857  Basecbs 15800  Posetcpo 16880  joincjn 16884  meetcmee 16885  Latclat 16985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-dm 5094  df-iota 5820  df-fv 5865  df-lat 16986
This theorem is referenced by:  latlej1  17000  latlej2  17001  latjle12  17002  latmle1  17016  latmle2  17017  latlem12  17018
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