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Theorem omllaw2N 39245
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 31604 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 39244 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
7 eqcom 2744 . . 3 ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (( 𝑋) 𝑌)))
8 omllat 39243 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ Lat)
983ad2ant1 1134 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
10 omlop 39242 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
111, 5opoccl 39195 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1210, 11sylan 580 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
13123adant3 1133 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
14 simp3 1139 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
151, 4latmcom 18508 . . . . . 6 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
169, 13, 14, 15syl3anc 1373 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
1716oveq2d 7447 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 (𝑌 ( 𝑋))))
1817eqeq2d 2748 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = (𝑋 (( 𝑋) 𝑌)) ↔ 𝑌 = (𝑋 (𝑌 ( 𝑋)))))
197, 18bitrid 283 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
206, 19sylibrd 259 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  occoc 17305  joincjn 18357  meetcmee 18358  Latclat 18476  OPcops 39173  OMLcoml 39176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-glb 18392  df-meet 18394  df-lat 18477  df-oposet 39177  df-ol 39179  df-oml 39180
This theorem is referenced by:  omllaw5N  39248  cmtcomlemN  39249  cmtbr3N  39255
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