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Theorem mod2ile 17458
Description: The weak direction of the modular law (e.g., pmod2iN 35923) 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 785 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝐾 ∈ Lat)
2 simplr3 1285 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑍𝐵)
3 simplr2 1283 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑌𝐵)
4 simplr1 1281 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑋𝐵)
52, 3, 43jca 1164 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍𝐵𝑌𝐵𝑋𝐵))
61, 5jca 509 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝐾 ∈ Lat ∧ (𝑍𝐵𝑌𝐵𝑋𝐵)))
7 simpr 479 . . . 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 17457 . . . 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 17427 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
151, 4, 3, 14syl3anc 1496 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 𝑌) = (𝑌 𝑋))
1615oveq1d 6919 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = ((𝑌 𝑋) 𝑍))
178, 11latmcl 17404 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
181, 3, 4, 17syl3anc 1496 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑋) ∈ 𝐵)
198, 10latjcom 17411 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌 𝑋) ∈ 𝐵𝑍𝐵) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
201, 18, 2, 19syl3anc 1496 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
2116, 20eqtrd 2860 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = (𝑍 (𝑌 𝑋)))
228, 10latjcom 17411 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑍 𝑌))
231, 3, 2, 22syl3anc 1496 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑍) = (𝑍 𝑌))
2423oveq2d 6920 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = (𝑋 (𝑍 𝑌)))
258, 10latjcl 17403 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
261, 2, 3, 25syl3anc 1496 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍 𝑌) ∈ 𝐵)
278, 11latmcom 17427 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑍 𝑌) ∈ 𝐵) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
281, 4, 26, 27syl3anc 1496 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
2924, 28eqtrd 2860 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = ((𝑍 𝑌) 𝑋))
3013, 21, 293brtr4d 4904 . 2 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)))
3130ex 403 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑋 → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166   class class class wbr 4872  cfv 6122  (class class class)co 6904  Basecbs 16221  lecple 16311  joincjn 17296  meetcmee 17297  Latclat 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-poset 17298  df-lub 17326  df-glb 17327  df-join 17328  df-meet 17329  df-lat 17398
This theorem is referenced by: (None)
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