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Theorem latledi 17297
Description: An ortholattice is distributive in one ordering direction. (ledi 28733 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 1228 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑋)
61, 2, 3latmle1 17284 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑋)
763adant3r2 1227 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑋)
81, 3latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
983adant3r3 1228 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
101, 3latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
11103adant3r2 1227 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
12 simpr1 1241 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
139, 11, 123jca 1151 . . . 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 581 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑍) 𝑋) ↔ ((𝑋 𝑌) (𝑋 𝑍)) 𝑋))
175, 7, 16mpbi2and 694 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) 𝑋)
181, 2, 3latmle2 17285 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
19183adant3r3 1228 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) 𝑌)
201, 2, 3latmle2 17285 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) 𝑍)
21203adant3r2 1227 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) 𝑍)
22 simpl 470 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
23 simpr2 1243 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
24 simpr3 1245 . . . 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 1491 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑌 ∧ (𝑋 𝑍) 𝑍) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑌 𝑍)))
2719, 21, 26mp2and 682 . 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 1483 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
301, 14latjcl 17259 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
31303adant3r1 1226 . . 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 1484 . 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 694 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157   class class class wbr 4851  cfv 6104  (class class class)co 6877  Basecbs 16071  lecple 16163  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 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254
This theorem is referenced by:  omlfh1N  35040
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