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Theorem mod2ile 17492
Description: The weak direction of the modular law (e.g., pmod2iN 36003) 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 757 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝐾 ∈ Lat)
2 simplr3 1236 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑍𝐵)
3 simplr2 1234 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑌𝐵)
4 simplr1 1232 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑋𝐵)
52, 3, 43jca 1119 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍𝐵𝑌𝐵𝑋𝐵))
61, 5jca 507 . . . 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 17491 . . . 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 17461 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
151, 4, 3, 14syl3anc 1439 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 𝑌) = (𝑌 𝑋))
1615oveq1d 6937 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = ((𝑌 𝑋) 𝑍))
178, 11latmcl 17438 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
181, 3, 4, 17syl3anc 1439 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑋) ∈ 𝐵)
198, 10latjcom 17445 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌 𝑋) ∈ 𝐵𝑍𝐵) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
201, 18, 2, 19syl3anc 1439 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
2116, 20eqtrd 2814 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = (𝑍 (𝑌 𝑋)))
228, 10latjcom 17445 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑍 𝑌))
231, 3, 2, 22syl3anc 1439 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑍) = (𝑍 𝑌))
2423oveq2d 6938 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = (𝑋 (𝑍 𝑌)))
258, 10latjcl 17437 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
261, 2, 3, 25syl3anc 1439 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍 𝑌) ∈ 𝐵)
278, 11latmcom 17461 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑍 𝑌) ∈ 𝐵) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
281, 4, 26, 27syl3anc 1439 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
2924, 28eqtrd 2814 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = ((𝑍 𝑌) 𝑋))
3013, 21, 293brtr4d 4918 . 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 1071   = wceq 1601  wcel 2107   class class class wbr 4886  cfv 6135  (class class class)co 6922  Basecbs 16255  lecple 16345  joincjn 17330  meetcmee 17331  Latclat 17431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-poset 17332  df-lub 17360  df-glb 17361  df-join 17362  df-meet 17363  df-lat 17432
This theorem is referenced by: (None)
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