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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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