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Theorem latledi 17983
Description: An ortholattice is distributive in one ordering direction. (ledi 29621 analog.) (Contributed by NM, 7-Nov-2011.)
Hypotheses
Ref Expression
latledi.b 𝐵 = (Base‘𝐾)
latledi.l = (le‘𝐾)
latledi.j = (join‘𝐾)
latledi.m = (meet‘𝐾)
Assertion
Ref Expression
latledi ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍)))

Proof of Theorem latledi
StepHypRef Expression
1 latledi.b . . . . 5 𝐵 = (Base‘𝐾)
2 latledi.l . . . . 5 = (le‘𝐾)
3 latledi.m . . . . 5 = (meet‘𝐾)
41, 2, 3latmle1 17970 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)
543adant3r3 1186 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑋)
61, 2, 3latmle1 17970 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑋)
763adant3r2 1185 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑋)
81, 3latmcl 17946 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
983adant3r3 1186 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
101, 3latmcl 17946 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
11103adant3r2 1185 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
12 simpr1 1196 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
139, 11, 123jca 1130 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵𝑋𝐵))
14 latledi.j . . . . 5 = (join‘𝐾)
151, 2, 14latjle12 17956 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵𝑋𝐵)) → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑍) 𝑋) ↔ ((𝑋 𝑌) (𝑋 𝑍)) 𝑋))
1613, 15syldan 594 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑍) 𝑋) ↔ ((𝑋 𝑌) (𝑋 𝑍)) 𝑋))
175, 7, 16mpbi2and 712 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) 𝑋)
181, 2, 3latmle2 17971 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
19183adant3r3 1186 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑌)
201, 2, 3latmle2 17971 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑍)
21203adant3r2 1185 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑍)
22 simpl 486 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
23 simpr2 1197 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
24 simpr3 1198 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
251, 2, 14latjlej12 17961 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ ((𝑋 𝑍) ∈ 𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑌 ∧ (𝑋 𝑍) 𝑍) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍)))
2622, 9, 23, 11, 24, 25syl122anc 1381 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑌 ∧ (𝑋 𝑍) 𝑍) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍)))
2719, 21, 26mp2and 699 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍))
281, 14latjcl 17945 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
2922, 9, 11, 28syl3anc 1373 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
301, 14latjcl 17945 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
31303adant3r1 1184 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
321, 2, 3latlem12 17972 . . 3 ((𝐾 ∈ Lat ∧ (((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵)) → ((((𝑋 𝑌) (𝑋 𝑍)) 𝑋 ∧ ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍)) ↔ ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍))))
3322, 29, 12, 31, 32syl13anc 1374 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) (𝑋 𝑍)) 𝑋 ∧ ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍)) ↔ ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍))))
3417, 27, 33mpbi2and 712 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110   class class class wbr 5053  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818  meetcmee 17819  Latclat 17937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-poset 17820  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-lat 17938
This theorem is referenced by:  omlfh1N  37009
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