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Theorem omlmod1i2N 36432
Description: Analogue of modular law atmod1i2 37031 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 1132 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OML)
2 simp23 1204 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
3 simp21 1202 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
4 simp22 1203 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 simp3l 1197 . . . . 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 36428 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍𝑋𝐶𝑍))
101, 3, 2, 9syl3anc 1367 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍𝑋𝐶𝑍))
115, 10mpd 15 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐶𝑍)
126, 8cmtcomN 36421 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑍𝐶𝑋))
131, 3, 2, 12syl3anc 1367 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋𝐶𝑍𝑍𝐶𝑋))
1411, 13mpbid 234 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑋)
15 simp3r 1198 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐶𝑍)
166, 8cmtcomN 36421 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍𝑍𝐶𝑌))
171, 4, 2, 16syl3anc 1367 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑌𝐶𝑍𝑍𝐶𝑌))
1815, 17mpbid 234 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑌)
19 omlmod.j . . . 4 = (join‘𝐾)
20 omlmod.m . . . 4 = (meet‘𝐾)
216, 19, 20, 8omlfh1N 36430 . . 3 ((𝐾 ∈ OML ∧ (𝑍𝐵𝑋𝐵𝑌𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
221, 2, 3, 4, 14, 18, 21syl132anc 1384 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
23 omllat 36414 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
24233ad2ant1 1129 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Lat)
256, 19latjcl 17640 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2624, 3, 4, 25syl3anc 1367 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑌) ∈ 𝐵)
276, 20latmcom 17664 . . 3 ((𝐾 ∈ Lat ∧ 𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
2824, 2, 26, 27syl3anc 1367 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
296, 7, 20latleeqm2 17669 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
3024, 3, 2, 29syl3anc 1367 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
315, 30mpbid 234 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑋) = 𝑋)
326, 20latmcom 17664 . . . 4 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) = (𝑌 𝑍))
3324, 2, 4, 32syl3anc 1367 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑌) = (𝑌 𝑍))
3431, 33oveq12d 7151 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → ((𝑍 𝑋) (𝑍 𝑌)) = (𝑋 (𝑌 𝑍)))
3522, 28, 343eqtr3rd 2864 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  lecple 16551  joincjn 17533  meetcmee 17534  Latclat 17634  cmccmtN 36345  OMLcoml 36347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-proset 17517  df-poset 17535  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-p0 17628  df-lat 17635  df-oposet 36348  df-cmtN 36349  df-ol 36350  df-oml 36351
This theorem is referenced by:  omlspjN  36433
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