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Theorem omllaw2N 38417
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 31093 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omllaw.b 𝐡 = (Baseβ€˜πΎ)
omllaw.l ≀ = (leβ€˜πΎ)
omllaw.j ∨ = (joinβ€˜πΎ)
omllaw.m ∧ = (meetβ€˜πΎ)
omllaw.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
omllaw2N ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 ∨ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ)) = π‘Œ))
Proof of Theorem omllaw2N
StepHypRef Expression
1 omllaw.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 omllaw.l . . 3 ≀ = (leβ€˜πΎ)
3 omllaw.j . . 3 ∨ = (joinβ€˜πΎ)
4 omllaw.m . . 3 ∧ = (meetβ€˜πΎ)
5 omllaw.o . . 3 βŠ₯ = (ocβ€˜πΎ)
61, 2, 3, 4, 5omllaw 38416 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))
7 eqcom 2739 . . 3 ((𝑋 ∨ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ)) = π‘Œ ↔ π‘Œ = (𝑋 ∨ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ)))
8 omllat 38415 . . . . . . 7 (𝐾 ∈ OML β†’ 𝐾 ∈ Lat)
983ad2ant1 1133 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
10 omlop 38414 . . . . . . . 8 (𝐾 ∈ OML β†’ 𝐾 ∈ OP)
111, 5opoccl 38367 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
1210, 11sylan 580 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
13123adant3 1132 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
14 simp3 1138 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
151, 4latmcom 18420 . . . . . 6 ((𝐾 ∈ Lat ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ) = (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))
169, 13, 14, 15syl3anc 1371 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ) = (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))
1716oveq2d 7427 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ)) = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1817eqeq2d 2743 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ = (𝑋 ∨ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ)) ↔ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))
197, 18bitrid 282 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∨ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ)) = π‘Œ ↔ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))
206, 19sylibrd 258 1 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 ∨ (( βŠ₯ β€˜π‘‹) ∧ π‘Œ)) = π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  lecple 17208  occoc 17209  joincjn 18268  meetcmee 18269  Latclat 18388  OPcops 38345  OMLcoml 38348
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-glb 18304  df-meet 18306  df-lat 18389  df-oposet 38349  df-ol 38351  df-oml 38352
This theorem is referenced by:  omllaw5N  38420  cmtcomlemN  38421  cmtbr3N  38427
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