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Theorem mod1ile 17465
Description: The weak direction of the modular law (e.g., pmod1i 35922, atmod1i1 35931) 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 783 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝐾 ∈ Lat)
2 simplr1 1279 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋𝐵)
3 simplr2 1281 . . . . 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 17420 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
81, 2, 3, 7syl3anc 1494 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 (𝑋 𝑌))
9 simpr 479 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 𝑍)
104, 6latjcl 17411 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
111, 2, 3, 10syl3anc 1494 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑋 𝑌) ∈ 𝐵)
12 simplr3 1283 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑍𝐵)
13 modle.m . . . . . 6 = (meet‘𝐾)
144, 5, 13latlem12 17438 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑋 (𝑋 𝑌) ∧ 𝑋 𝑍) ↔ 𝑋 ((𝑋 𝑌) 𝑍)))
151, 2, 11, 12, 14syl13anc 1495 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 (𝑋 𝑌) ∧ 𝑋 𝑍) ↔ 𝑋 ((𝑋 𝑌) 𝑍)))
168, 9, 15mpbi2and 703 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → 𝑋 ((𝑋 𝑌) 𝑍))
174, 5, 6, 13latmlej12 17451 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌𝐵𝑍𝐵𝑋𝐵)) → (𝑌 𝑍) (𝑋 𝑌))
181, 3, 12, 2, 17syl13anc 1495 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) (𝑋 𝑌))
194, 5, 13latmle2 17437 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) 𝑍)
201, 3, 12, 19syl3anc 1494 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) 𝑍)
214, 13latmcl 17412 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
221, 3, 12, 21syl3anc 1494 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) ∈ 𝐵)
234, 5, 13latlem12 17438 . . . . 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 1495 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (((𝑌 𝑍) (𝑋 𝑌) ∧ (𝑌 𝑍) 𝑍) ↔ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)))
2518, 20, 24mpbi2and 703 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑌 𝑍) ((𝑋 𝑌) 𝑍))
264, 13latmcl 17412 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
271, 11, 12, 26syl3anc 1494 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
284, 5, 6latjle12 17422 . . . 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 1495 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → ((𝑋 ((𝑋 𝑌) 𝑍) ∧ (𝑌 𝑍) ((𝑋 𝑌) 𝑍)) ↔ (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
3016, 25, 29mpbi2and 703 . 2 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑍) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
3130ex 403 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  meetcmee 17305  Latclat 17405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-poset 17306  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-lat 17406
This theorem is referenced by:  mod2ile  17466  hlmod1i  35930
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