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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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