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Theorem omllaw2N 35026
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 28778 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 35025 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
7 eqcom 2820 . . 3 ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (( 𝑋) 𝑌)))
8 omllat 35024 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ Lat)
983ad2ant1 1156 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
10 omlop 35023 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
111, 5opoccl 34976 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1210, 11sylan 571 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
13123adant3 1155 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
14 simp3 1161 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
151, 4latmcom 17283 . . . . . 6 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
169, 13, 14, 15syl3anc 1483 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) = (𝑌 ( 𝑋)))
1716oveq2d 6893 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 (𝑌 ( 𝑋))))
1817eqeq2d 2823 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = (𝑋 (( 𝑋) 𝑌)) ↔ 𝑌 = (𝑋 (𝑌 ( 𝑋)))))
197, 18syl5bb 274 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
206, 19sylibrd 250 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1100   = wceq 1637  wcel 2157   class class class wbr 4851  cfv 6104  (class class class)co 6877  Basecbs 16071  lecple 16163  occoc 16164  joincjn 17152  meetcmee 17153  Latclat 17253  OPcops 34954  OMLcoml 34957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-glb 17183  df-meet 17185  df-lat 17254  df-oposet 34958  df-ol 34960  df-oml 34961
This theorem is referenced by:  omllaw5N  35029  cmtcomlemN  35030  cmtbr3N  35036
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