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Theorem omlmod1i2N 34024
 Description: Analogue of modular law atmod1i2 34622 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 1059 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OML)
2 simp23 1094 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
3 simp21 1092 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
4 simp22 1093 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 simp3l 1087 . . . . 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 34020 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍𝑋𝐶𝑍))
101, 3, 2, 9syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍𝑋𝐶𝑍))
115, 10mpd 15 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐶𝑍)
126, 8cmtcomN 34013 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑍𝐶𝑋))
131, 3, 2, 12syl3anc 1323 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋𝐶𝑍𝑍𝐶𝑋))
1411, 13mpbid 222 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑋)
15 simp3r 1088 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐶𝑍)
166, 8cmtcomN 34013 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍𝑍𝐶𝑌))
171, 4, 2, 16syl3anc 1323 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑌𝐶𝑍𝑍𝐶𝑌))
1815, 17mpbid 222 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑌)
19 omlmod.j . . . 4 = (join‘𝐾)
20 omlmod.m . . . 4 = (meet‘𝐾)
216, 19, 20, 8omlfh1N 34022 . . 3 ((𝐾 ∈ OML ∧ (𝑍𝐵𝑋𝐵𝑌𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
221, 2, 3, 4, 14, 18, 21syl132anc 1341 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
23 omllat 34006 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
24233ad2ant1 1080 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Lat)
256, 19latjcl 16972 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2624, 3, 4, 25syl3anc 1323 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑌) ∈ 𝐵)
276, 20latmcom 16996 . . 3 ((𝐾 ∈ Lat ∧ 𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
2824, 2, 26, 27syl3anc 1323 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
296, 7, 20latleeqm2 17001 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
3024, 3, 2, 29syl3anc 1323 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
315, 30mpbid 222 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑋) = 𝑋)
326, 20latmcom 16996 . . . 4 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) = (𝑌 𝑍))
3324, 2, 4, 32syl3anc 1323 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑌) = (𝑌 𝑍))
3431, 33oveq12d 6622 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → ((𝑍 𝑋) (𝑍 𝑌)) = (𝑋 (𝑌 𝑍)))
3522, 28, 343eqtr3rd 2664 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   class class class wbr 4613  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  meetcmee 16866  Latclat 16966  cmccmtN 33937  OMLcoml 33939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-oposet 33940  df-cmtN 33941  df-ol 33942  df-oml 33943 This theorem is referenced by:  omlspjN  34025
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