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Theorem omlmod1i2N 38859
Description: Analogue of modular law atmod1i2 39459 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlmod.b 𝐵 = (Base‘𝐾)
omlmod.l = (le‘𝐾)
omlmod.j = (join‘𝐾)
omlmod.m = (meet‘𝐾)
omlmod.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
omlmod1i2N ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))

Proof of Theorem omlmod1i2N
StepHypRef Expression
1 simp1 1133 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OML)
2 simp23 1205 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
3 simp21 1203 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
4 simp22 1204 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 simp3l 1198 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋 𝑍)
6 omlmod.b . . . . . . 7 𝐵 = (Base‘𝐾)
7 omlmod.l . . . . . . 7 = (le‘𝐾)
8 omlmod.c . . . . . . 7 𝐶 = (cm‘𝐾)
96, 7, 8lecmtN 38855 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍𝑋𝐶𝑍))
101, 3, 2, 9syl3anc 1368 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍𝑋𝐶𝑍))
115, 10mpd 15 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐶𝑍)
126, 8cmtcomN 38848 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑍𝐶𝑋))
131, 3, 2, 12syl3anc 1368 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋𝐶𝑍𝑍𝐶𝑋))
1411, 13mpbid 231 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑋)
15 simp3r 1199 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐶𝑍)
166, 8cmtcomN 38848 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍𝑍𝐶𝑌))
171, 4, 2, 16syl3anc 1368 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑌𝐶𝑍𝑍𝐶𝑌))
1815, 17mpbid 231 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑌)
19 omlmod.j . . . 4 = (join‘𝐾)
20 omlmod.m . . . 4 = (meet‘𝐾)
216, 19, 20, 8omlfh1N 38857 . . 3 ((𝐾 ∈ OML ∧ (𝑍𝐵𝑋𝐵𝑌𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
221, 2, 3, 4, 14, 18, 21syl132anc 1385 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
23 omllat 38841 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
24233ad2ant1 1130 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Lat)
256, 19latjcl 18434 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2624, 3, 4, 25syl3anc 1368 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑌) ∈ 𝐵)
276, 20latmcom 18458 . . 3 ((𝐾 ∈ Lat ∧ 𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
2824, 2, 26, 27syl3anc 1368 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
296, 7, 20latleeqm2 18463 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
3024, 3, 2, 29syl3anc 1368 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
315, 30mpbid 231 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑋) = 𝑋)
326, 20latmcom 18458 . . . 4 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) = (𝑌 𝑍))
3324, 2, 4, 32syl3anc 1368 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑌) = (𝑌 𝑍))
3431, 33oveq12d 7437 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → ((𝑍 𝑋) (𝑍 𝑌)) = (𝑋 (𝑌 𝑍)))
3522, 28, 343eqtr3rd 2774 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  meetcmee 18307  Latclat 18426  cmccmtN 38772  OMLcoml 38774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18290  df-poset 18308  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-lat 18427  df-oposet 38775  df-cmtN 38776  df-ol 38777  df-oml 38778
This theorem is referenced by:  omlspjN  38860
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