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Theorem omllaw3 38419
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 30957 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 7420 . . . . . 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 38414 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OL)
4 omllaw3.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
5 eqid 2731 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
6 omllaw3.z . . . . . . . . 9 0 = (0.‘𝐾)
74, 5, 6olj01 38399 . . . . . . . 8 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
83, 7sylan 579 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
983adant3 1131 . . . . . 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 2772 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1211adantrl 713 . . 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 38417 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋)))))
1716imp 406 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1817adantrr 714 . . 3 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1912, 18eqtr4d 2774 . 2 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑋 = 𝑌)
2019ex 412 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17149  lecple 17209  occoc 17210  joincjn 18269  meetcmee 18270  0.cp0 18381  OLcol 38348  OMLcoml 38349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-proset 18253  df-poset 18271  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18390  df-oposet 38350  df-ol 38352  df-oml 38353
This theorem is referenced by:  omlfh1N  38432  atlatmstc  38493
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