Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omllaw3 Structured version   Visualization version   GIF version

Theorem omllaw3 33550
 Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 27679 analog.) (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
omllaw3.b 𝐵 = (Base‘𝐾)
omllaw3.l = (le‘𝐾)
omllaw3.m = (meet‘𝐾)
omllaw3.o = (oc‘𝐾)
omllaw3.z 0 = (0.‘𝐾)
Assertion
Ref Expression
omllaw3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))

Proof of Theorem omllaw3
StepHypRef Expression
1 oveq2 6557 . . . . . 6 ((𝑌 ( 𝑋)) = 0 → (𝑋(join‘𝐾)(𝑌 ( 𝑋))) = (𝑋(join‘𝐾) 0 ))
21adantl 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → (𝑋(join‘𝐾)(𝑌 ( 𝑋))) = (𝑋(join‘𝐾) 0 ))
3 omlol 33545 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OL)
4 omllaw3.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
5 eqid 2610 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
6 omllaw3.z . . . . . . . . 9 0 = (0.‘𝐾)
74, 5, 6olj01 33530 . . . . . . . 8 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
83, 7sylan 487 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
983adant3 1074 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
109adantr 480 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → (𝑋(join‘𝐾) 0 ) = 𝑋)
112, 10eqtr2d 2645 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1211adantrl 748 . . 3 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
13 omllaw3.l . . . . . 6 = (le‘𝐾)
14 omllaw3.m . . . . . 6 = (meet‘𝐾)
15 omllaw3.o . . . . . 6 = (oc‘𝐾)
164, 13, 5, 14, 15omllaw 33548 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋)))))
1716imp 444 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1817adantrr 749 . . 3 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1912, 18eqtr4d 2647 . 2 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑋 = 𝑌)
2019ex 449 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  occoc 15776  joincjn 16767  meetcmee 16768  0.cp0 16860  OLcol 33479  OMLcoml 33480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-oposet 33481  df-ol 33483  df-oml 33484 This theorem is referenced by:  omlfh1N  33563  atlatmstc  33624
 Copyright terms: Public domain W3C validator