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Theorem latledi 17297
 Description: An ortholattice is distributive in one ordering direction. (ledi 28739 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 17284 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)
543adant3r3 1199 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑋)
61, 2, 3latmle1 17284 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑋)
763adant3r2 1198 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑋)
81, 3latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
983adant3r3 1199 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
101, 3latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
11103adant3r2 1198 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
12 simpr1 1233 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
139, 11, 123jca 1122 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵𝑋𝐵))
14 latledi.j . . . . 5 = (join‘𝐾)
151, 2, 14latjle12 17270 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵𝑋𝐵)) → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑍) 𝑋) ↔ ((𝑋 𝑌) (𝑋 𝑍)) 𝑋))
1613, 15syldan 579 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑍) 𝑋) ↔ ((𝑋 𝑌) (𝑋 𝑍)) 𝑋))
175, 7, 16mpbi2and 691 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) 𝑋)
181, 2, 3latmle2 17285 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
19183adant3r3 1199 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑌)
201, 2, 3latmle2 17285 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑍)
21203adant3r2 1198 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑍)
22 simpl 468 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
23 simpr2 1235 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
24 simpr3 1237 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
251, 2, 14latjlej12 17275 . . . 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 1485 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑌 ∧ (𝑋 𝑍) 𝑍) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍)))
2719, 21, 26mp2and 679 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍))
281, 14latjcl 17259 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
2922, 9, 11, 28syl3anc 1476 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
301, 14latjcl 17259 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
31303adant3r1 1197 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
321, 2, 3latlem12 17286 . . 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 1478 . 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 691 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254 This theorem is referenced by:  omlfh1N  35067
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