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Theorem omllaw2N 36537
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 29368 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 36536 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
7 eqcom 2805 . . 3 ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (( 𝑋) 𝑌)))
8 omllat 36535 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ Lat)
983ad2ant1 1130 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
10 omlop 36534 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
111, 5opoccl 36487 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1210, 11sylan 583 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
13123adant3 1129 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
14 simp3 1135 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
151, 4latmcom 17677 . . . . . 6 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
169, 13, 14, 15syl3anc 1368 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
1716oveq2d 7151 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 (𝑌 ( 𝑋))))
1817eqeq2d 2809 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = (𝑋 (( 𝑋) 𝑌)) ↔ 𝑌 = (𝑋 (𝑌 ( 𝑋)))))
197, 18syl5bb 286 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
206, 19sylibrd 262 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2111   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  occoc 16565  joincjn 17546  meetcmee 17547  Latclat 17647  OPcops 36465  OMLcoml 36468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-glb 17577  df-meet 17579  df-lat 17648  df-oposet 36469  df-ol 36471  df-oml 36472
This theorem is referenced by:  omllaw5N  36540  cmtcomlemN  36541  cmtbr3N  36547
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