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Theorem omllaw5N 35052
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 28808 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 1130 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
2 simp2 1131 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 omllat 35047 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
4 omllaw5.b . . . . 5 𝐵 = (Base‘𝐾)
5 omllaw5.j . . . . 5 = (join‘𝐾)
64, 5latjcl 17255 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
73, 6syl3an1 1166 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
81, 2, 73jca 1122 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵))
9 eqid 2771 . . . 4 (le‘𝐾) = (le‘𝐾)
104, 9, 5latlej1 17264 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
113, 10syl3an1 1166 . 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 35049 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑋(le‘𝐾)(𝑋 𝑌) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌)))
158, 11, 14sylc 65 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6029  (class class class)co 6792  Basecbs 16060  lecple 16152  occoc 16153  joincjn 17148  meetcmee 17149  Latclat 17249  OMLcoml 34980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6753  df-ov 6795  df-oprab 6796  df-lub 17178  df-glb 17179  df-join 17180  df-meet 17181  df-lat 17250  df-oposet 34981  df-ol 34983  df-oml 34984
This theorem is referenced by: (None)
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