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Theorem omllaw5N 33376
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 27690 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omllaw5.b 𝐵 = (Base‘𝐾)
omllaw5.j = (join‘𝐾)
omllaw5.m = (meet‘𝐾)
omllaw5.o = (oc‘𝐾)
Assertion
Ref Expression
omllaw5N ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌))

Proof of Theorem omllaw5N
StepHypRef Expression
1 simp1 1053 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
2 simp2 1054 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 omllat 33371 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
4 omllaw5.b . . . . 5 𝐵 = (Base‘𝐾)
5 omllaw5.j . . . . 5 = (join‘𝐾)
64, 5latjcl 16823 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
73, 6syl3an1 1350 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
81, 2, 73jca 1234 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵))
9 eqid 2609 . . . 4 (le‘𝐾) = (le‘𝐾)
104, 9, 5latlej1 16832 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
113, 10syl3an1 1350 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
12 omllaw5.m . . 3 = (meet‘𝐾)
13 omllaw5.o . . 3 = (oc‘𝐾)
144, 9, 5, 12, 13omllaw2N 33373 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑋(le‘𝐾)(𝑋 𝑌) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌)))
158, 11, 14sylc 62 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  occoc 15725  joincjn 16716  meetcmee 16717  Latclat 16817  OMLcoml 33304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-lat 16818  df-oposet 33305  df-ol 33307  df-oml 33308
This theorem is referenced by: (None)
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