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Theorem mod2ile 17100
 Description: The weak direction of the modular law (e.g., pmod2iN 34961) 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
mod2ile ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑋 → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍))))

Proof of Theorem mod2ile
StepHypRef Expression
1 simpll 790 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝐾 ∈ Lat)
2 simplr3 1104 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑍𝐵)
3 simplr2 1103 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑌𝐵)
4 simplr1 1102 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑋𝐵)
52, 3, 43jca 1241 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍𝐵𝑌𝐵𝑋𝐵))
61, 5jca 554 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝐾 ∈ Lat ∧ (𝑍𝐵𝑌𝐵𝑋𝐵)))
7 simpr 477 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑍 𝑋)
8 modle.b . . . . 5 𝐵 = (Base‘𝐾)
9 modle.l . . . . 5 = (le‘𝐾)
10 modle.j . . . . 5 = (join‘𝐾)
11 modle.m . . . . 5 = (meet‘𝐾)
128, 9, 10, 11mod1ile 17099 . . . 4 ((𝐾 ∈ Lat ∧ (𝑍𝐵𝑌𝐵𝑋𝐵)) → (𝑍 𝑋 → (𝑍 (𝑌 𝑋)) ((𝑍 𝑌) 𝑋)))
136, 7, 12sylc 65 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍 (𝑌 𝑋)) ((𝑍 𝑌) 𝑋))
148, 11latmcom 17069 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
151, 4, 3, 14syl3anc 1325 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 𝑌) = (𝑌 𝑋))
1615oveq1d 6662 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = ((𝑌 𝑋) 𝑍))
178, 11latmcl 17046 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
181, 3, 4, 17syl3anc 1325 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑋) ∈ 𝐵)
198, 10latjcom 17053 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌 𝑋) ∈ 𝐵𝑍𝐵) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
201, 18, 2, 19syl3anc 1325 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
2116, 20eqtrd 2655 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = (𝑍 (𝑌 𝑋)))
228, 10latjcom 17053 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑍 𝑌))
231, 3, 2, 22syl3anc 1325 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑍) = (𝑍 𝑌))
2423oveq2d 6663 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = (𝑋 (𝑍 𝑌)))
258, 10latjcl 17045 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
261, 2, 3, 25syl3anc 1325 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍 𝑌) ∈ 𝐵)
278, 11latmcom 17069 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑍 𝑌) ∈ 𝐵) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
281, 4, 26, 27syl3anc 1325 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
2924, 28eqtrd 2655 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = ((𝑍 𝑌) 𝑋))
3013, 21, 293brtr4d 4683 . 2 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)))
3130ex 450 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑋 → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1482   ∈ wcel 1989   class class class wbr 4651  ‘cfv 5886  (class class class)co 6647  Basecbs 15851  lecple 15942  joincjn 16938  meetcmee 16939  Latclat 17039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-poset 16940  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-lat 17040 This theorem is referenced by: (None)
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