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Theorem mod2ile 17307
Description: The weak direction of the modular law (e.g., pmod2iN 35638) 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 807 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝐾 ∈ Lat)
2 simplr3 1265 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑍𝐵)
3 simplr2 1263 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑌𝐵)
4 simplr1 1261 . . . . . 6 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → 𝑋𝐵)
52, 3, 43jca 1123 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍𝐵𝑌𝐵𝑋𝐵))
61, 5jca 555 . . . 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 17306 . . . 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 17276 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
151, 4, 3, 14syl3anc 1477 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 𝑌) = (𝑌 𝑋))
1615oveq1d 6828 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = ((𝑌 𝑋) 𝑍))
178, 11latmcl 17253 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
181, 3, 4, 17syl3anc 1477 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑋) ∈ 𝐵)
198, 10latjcom 17260 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌 𝑋) ∈ 𝐵𝑍𝐵) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
201, 18, 2, 19syl3anc 1477 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑌 𝑋) 𝑍) = (𝑍 (𝑌 𝑋)))
2116, 20eqtrd 2794 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) = (𝑍 (𝑌 𝑋)))
228, 10latjcom 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑍 𝑌))
231, 3, 2, 22syl3anc 1477 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑌 𝑍) = (𝑍 𝑌))
2423oveq2d 6829 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = (𝑋 (𝑍 𝑌)))
258, 10latjcl 17252 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
261, 2, 3, 25syl3anc 1477 . . . . 5 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑍 𝑌) ∈ 𝐵)
278, 11latmcom 17276 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑍 𝑌) ∈ 𝐵) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
281, 4, 26, 27syl3anc 1477 . . . 4 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑍 𝑌)) = ((𝑍 𝑌) 𝑋))
2924, 28eqtrd 2794 . . 3 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → (𝑋 (𝑌 𝑍)) = ((𝑍 𝑌) 𝑋))
3013, 21, 293brtr4d 4836 . 2 (((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑍 𝑋) → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)))
3130ex 449 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑋 → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Latclat 17246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-poset 17147  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-lat 17247
This theorem is referenced by: (None)
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