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Theorem omllaw5N 35027
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 28800 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 1159 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
2 simp2 1160 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 omllat 35022 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
4 omllaw5.b . . . . 5 𝐵 = (Base‘𝐾)
5 omllaw5.j . . . . 5 = (join‘𝐾)
64, 5latjcl 17256 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
73, 6syl3an1 1195 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
81, 2, 73jca 1151 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵))
9 eqid 2806 . . . 4 (le‘𝐾) = (le‘𝐾)
104, 9, 5latlej1 17265 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
113, 10syl3an1 1195 . 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 35024 . 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 1100   = wceq 1637  wcel 2156   class class class wbr 4844  cfv 6101  (class class class)co 6874  Basecbs 16068  lecple 16160  occoc 16161  joincjn 17149  meetcmee 17150  Latclat 17250  OMLcoml 34955
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 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
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 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-lub 17179  df-glb 17180  df-join 17181  df-meet 17182  df-lat 17251  df-oposet 34956  df-ol 34958  df-oml 34959
This theorem is referenced by: (None)
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