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Theorem mod1ile 17710
 Description: The weak direction of the modular law (e.g., pmod1i 36870, atmod1i1 36879) 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 763 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝐾 ∈ Lat)
2 simplr1 1209 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋𝐵)
3 simplr2 1210 . . . . 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 17665 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
81, 2, 3, 7syl3anc 1365 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 (𝑋 𝑌))
9 simpr 485 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 𝑍)
104, 6latjcl 17656 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
111, 2, 3, 10syl3anc 1365 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑋 𝑌) ∈ 𝐵)
12 simplr3 1211 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑍𝐵)
13 modle.m . . . . . 6 = (meet‘𝐾)
144, 5, 13latlem12 17683 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑋 (𝑋 𝑌) ∧ 𝑋 𝑍) ↔ 𝑋 ((𝑋 𝑌) 𝑍)))
151, 2, 11, 12, 14syl13anc 1366 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 (𝑋 𝑌) ∧ 𝑋 𝑍) ↔ 𝑋 ((𝑋 𝑌) 𝑍)))
168, 9, 15mpbi2and 708 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 ((𝑋 𝑌) 𝑍))
174, 5, 6, 13latmlej12 17696 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌𝐵𝑍𝐵𝑋𝐵)) → (𝑌 𝑍) (𝑋 𝑌))
181, 3, 12, 2, 17syl13anc 1366 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) (𝑋 𝑌))
194, 5, 13latmle2 17682 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) 𝑍)
201, 3, 12, 19syl3anc 1365 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) 𝑍)
214, 13latmcl 17657 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
221, 3, 12, 21syl3anc 1365 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) ∈ 𝐵)
234, 5, 13latlem12 17683 . . . . 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 1366 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (((𝑌 𝑍) (𝑋 𝑌) ∧ (𝑌 𝑍) 𝑍) ↔ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)))
2518, 20, 24mpbi2and 708 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) ((𝑋 𝑌) 𝑍))
264, 13latmcl 17657 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
271, 11, 12, 26syl3anc 1365 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
284, 5, 6latjle12 17667 . . . 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 1366 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 ((𝑋 𝑌) 𝑍) ∧ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)) ↔ (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
3016, 25, 29mpbi2and 708 . 2 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
3130ex 413 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   class class class wbr 5063  ‘cfv 6354  (class class class)co 7150  Basecbs 16478  lecple 16567  joincjn 17549  meetcmee 17550  Latclat 17650 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-poset 17551  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-lat 17651 This theorem is referenced by:  mod2ile  17711  hlmod1i  36878
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