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Theorem omllaw2N 34008
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 28290 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omllaw.b 𝐵 = (Base‘𝐾)
omllaw.l = (le‘𝐾)
omllaw.j = (join‘𝐾)
omllaw.m = (meet‘𝐾)
omllaw.o = (oc‘𝐾)
Assertion
Ref Expression
omllaw2N ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))

Proof of Theorem omllaw2N
StepHypRef Expression
1 omllaw.b . . 3 𝐵 = (Base‘𝐾)
2 omllaw.l . . 3 = (le‘𝐾)
3 omllaw.j . . 3 = (join‘𝐾)
4 omllaw.m . . 3 = (meet‘𝐾)
5 omllaw.o . . 3 = (oc‘𝐾)
61, 2, 3, 4, 5omllaw 34007 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
7 eqcom 2628 . . 3 ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (( 𝑋) 𝑌)))
8 omllat 34006 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ Lat)
983ad2ant1 1080 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
10 omlop 34005 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
111, 5opoccl 33958 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1210, 11sylan 488 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
13123adant3 1079 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
14 simp3 1061 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
151, 4latmcom 16996 . . . . . 6 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
169, 13, 14, 15syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
1716oveq2d 6620 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 (𝑌 ( 𝑋))))
1817eqeq2d 2631 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = (𝑋 (( 𝑋) 𝑌)) ↔ 𝑌 = (𝑋 (𝑌 ( 𝑋)))))
197, 18syl5bb 272 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
206, 19sylibrd 249 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  occoc 15870  joincjn 16865  meetcmee 16866  Latclat 16966  OPcops 33936  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-glb 16896  df-meet 16898  df-lat 16967  df-oposet 33940  df-ol 33942  df-oml 33943
This theorem is referenced by:  omllaw5N  34011  cmtcomlemN  34012  cmtbr3N  34018
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