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Theorem latledi 17701
Description: An ortholattice is distributive in one ordering direction. (ledi 29319 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 17688 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)
543adant3r3 1180 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑋)
61, 2, 3latmle1 17688 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑋)
763adant3r2 1179 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑋)
81, 3latmcl 17664 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
983adant3r3 1180 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
101, 3latmcl 17664 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
11103adant3r2 1179 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
12 simpr1 1190 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
139, 11, 123jca 1124 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵𝑋𝐵))
14 latledi.j . . . . 5 = (join‘𝐾)
151, 2, 14latjle12 17674 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵𝑋𝐵)) → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑍) 𝑋) ↔ ((𝑋 𝑌) (𝑋 𝑍)) 𝑋))
1613, 15syldan 593 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑍) 𝑋) ↔ ((𝑋 𝑌) (𝑋 𝑍)) 𝑋))
175, 7, 16mpbi2and 710 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) 𝑋)
181, 2, 3latmle2 17689 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
19183adant3r3 1180 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑌)
201, 2, 3latmle2 17689 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑍)
21203adant3r2 1179 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑍)
22 simpl 485 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
23 simpr2 1191 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
24 simpr3 1192 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
251, 2, 14latjlej12 17679 . . . 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 1375 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑌 ∧ (𝑋 𝑍) 𝑍) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍)))
2719, 21, 26mp2and 697 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍))
281, 14latjcl 17663 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
2922, 9, 11, 28syl3anc 1367 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
301, 14latjcl 17663 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
31303adant3r1 1178 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
321, 2, 3latlem12 17690 . . 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 1368 . 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 710 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114   class class class wbr 5068  cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  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-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463
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-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-poset 17558  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-lat 17658
This theorem is referenced by:  omlfh1N  36396
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