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Theorem latabs1 17700
 Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 29305 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
latabs1.b 𝐵 = (Base‘𝐾)
latabs1.j = (join‘𝐾)
latabs1.m = (meet‘𝐾)
Assertion
Ref Expression
latabs1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)

Proof of Theorem latabs1
StepHypRef Expression
1 latabs1.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2824 . . 3 (le‘𝐾) = (le‘𝐾)
3 latabs1.m . . 3 = (meet‘𝐾)
41, 2, 3latmle1 17689 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
51, 3latmcl 17665 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
6 latabs1.j . . . . 5 = (join‘𝐾)
71, 2, 6latleeqj2 17677 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑋𝐵) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 (𝑋 𝑌)) = 𝑋))
873com23 1123 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 (𝑋 𝑌)) = 𝑋))
95, 8syld3an3 1406 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 (𝑋 𝑌)) = 𝑋))
104, 9mpbid 235 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   class class class wbr 5053  ‘cfv 6344  (class class class)co 7150  Basecbs 16486  lecple 16575  joincjn 17557  meetcmee 17558  Latclat 17658 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17541  df-poset 17559  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-lat 17659 This theorem is referenced by:  latdisdlem  17802  cvrexchlem  36661
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