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Theorem mod1ile 17037
Description: The weak direction of the modular law (e.g., pmod1i 34649, atmod1i1 34658) that holds in any lattice. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
modle.b 𝐵 = (Base‘𝐾)
modle.l = (le‘𝐾)
modle.j = (join‘𝐾)
modle.m = (meet‘𝐾)
Assertion
Ref Expression
mod1ile ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))

Proof of Theorem mod1ile
StepHypRef Expression
1 simpll 789 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝐾 ∈ Lat)
2 simplr1 1101 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋𝐵)
3 simplr2 1102 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑌𝐵)
4 modle.b . . . . . 6 𝐵 = (Base‘𝐾)
5 modle.l . . . . . 6 = (le‘𝐾)
6 modle.j . . . . . 6 = (join‘𝐾)
74, 5, 6latlej1 16992 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
81, 2, 3, 7syl3anc 1323 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 (𝑋 𝑌))
9 simpr 477 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 𝑍)
104, 6latjcl 16983 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
111, 2, 3, 10syl3anc 1323 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑋 𝑌) ∈ 𝐵)
12 simplr3 1103 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑍𝐵)
13 modle.m . . . . . 6 = (meet‘𝐾)
144, 5, 13latlem12 17010 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑋 (𝑋 𝑌) ∧ 𝑋 𝑍) ↔ 𝑋 ((𝑋 𝑌) 𝑍)))
151, 2, 11, 12, 14syl13anc 1325 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 (𝑋 𝑌) ∧ 𝑋 𝑍) ↔ 𝑋 ((𝑋 𝑌) 𝑍)))
168, 9, 15mpbi2and 955 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 ((𝑋 𝑌) 𝑍))
174, 5, 6, 13latmlej12 17023 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌𝐵𝑍𝐵𝑋𝐵)) → (𝑌 𝑍) (𝑋 𝑌))
181, 3, 12, 2, 17syl13anc 1325 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) (𝑋 𝑌))
194, 5, 13latmle2 17009 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) 𝑍)
201, 3, 12, 19syl3anc 1323 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) 𝑍)
214, 13latmcl 16984 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
221, 3, 12, 21syl3anc 1323 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) ∈ 𝐵)
234, 5, 13latlem12 17010 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑌 𝑍) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → (((𝑌 𝑍) (𝑋 𝑌) ∧ (𝑌 𝑍) 𝑍) ↔ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)))
241, 22, 11, 12, 23syl13anc 1325 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (((𝑌 𝑍) (𝑋 𝑌) ∧ (𝑌 𝑍) 𝑍) ↔ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)))
2518, 20, 24mpbi2and 955 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) ((𝑋 𝑌) 𝑍))
264, 13latmcl 16984 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
271, 11, 12, 26syl3anc 1323 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
284, 5, 6latjle12 16994 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵 ∧ ((𝑋 𝑌) 𝑍) ∈ 𝐵)) → ((𝑋 ((𝑋 𝑌) 𝑍) ∧ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)) ↔ (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
291, 2, 22, 27, 28syl13anc 1325 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 ((𝑋 𝑌) 𝑍) ∧ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)) ↔ (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
3016, 25, 29mpbi2and 955 . 2 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
3130ex 450 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  joincjn 16876  meetcmee 16877  Latclat 16977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-poset 16878  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-lat 16978
This theorem is referenced by:  mod2ile  17038  hlmod1i  34657
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