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Theorem omllaw5N 36823
 Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 29495 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 1133 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
2 simp2 1134 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 omllat 36818 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
4 omllaw5.b . . . . 5 𝐵 = (Base‘𝐾)
5 omllaw5.j . . . . 5 = (join‘𝐾)
64, 5latjcl 17727 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
73, 6syl3an1 1160 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
81, 2, 73jca 1125 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵))
9 eqid 2758 . . . 4 (le‘𝐾) = (le‘𝐾)
104, 9, 5latlej1 17736 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
113, 10syl3an1 1160 . 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 36820 . 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 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5032  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  lecple 16630  occoc 16631  joincjn 17620  meetcmee 17621  Latclat 17721  OMLcoml 36751 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-lub 17650  df-glb 17651  df-join 17652  df-meet 17653  df-lat 17722  df-oposet 36752  df-ol 36754  df-oml 36755 This theorem is referenced by: (None)
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