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Theorem latlem 17030
Description: Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
latlem.b 𝐵 = (Base‘𝐾)
latlem.j = (join‘𝐾)
latlem.m = (meet‘𝐾)
Assertion
Ref Expression
latlem ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵))

Proof of Theorem latlem
StepHypRef Expression
1 latlem.b . . 3 𝐵 = (Base‘𝐾)
2 latlem.j . . 3 = (join‘𝐾)
3 simp1 1059 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
4 simp2 1060 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
5 simp3 1061 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
6 opelxpi 5138 . . . . 5 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
763adant1 1077 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
8 latlem.m . . . . . . 7 = (meet‘𝐾)
91, 2, 8islat 17028 . . . . . 6 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
10 simprl 793 . . . . . 6 ((𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))) → dom = (𝐵 × 𝐵))
119, 10sylbi 207 . . . . 5 (𝐾 ∈ Lat → dom = (𝐵 × 𝐵))
12113ad2ant1 1080 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → dom = (𝐵 × 𝐵))
137, 12eleqtrrd 2702 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ dom )
141, 2, 3, 4, 5, 13joincl 16987 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
15 simprr 795 . . . . . 6 ((𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))) → dom = (𝐵 × 𝐵))
169, 15sylbi 207 . . . . 5 (𝐾 ∈ Lat → dom = (𝐵 × 𝐵))
17163ad2ant1 1080 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → dom = (𝐵 × 𝐵))
187, 17eleqtrrd 2702 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ dom )
191, 8, 3, 4, 5, 18meetcl 17001 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2014, 19jca 554 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  cop 4174   × cxp 5102  dom cdm 5104  cfv 5876  (class class class)co 6635  Basecbs 15838  Posetcpo 16921  joincjn 16925  meetcmee 16926  Latclat 17026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-lat 17027
This theorem is referenced by:  latjcl  17032  latmcl  17033
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