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Theorem omllaw3 36375
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 29207 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 7158 . . . . . 6 ((𝑌 ( 𝑋)) = 0 → (𝑋(join‘𝐾)(𝑌 ( 𝑋))) = (𝑋(join‘𝐾) 0 ))
21adantl 484 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → (𝑋(join‘𝐾)(𝑌 ( 𝑋))) = (𝑋(join‘𝐾) 0 ))
3 omlol 36370 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OL)
4 omllaw3.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
5 eqid 2821 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
6 omllaw3.z . . . . . . . . 9 0 = (0.‘𝐾)
74, 5, 6olj01 36355 . . . . . . . 8 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
83, 7sylan 582 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
983adant3 1128 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
109adantr 483 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → (𝑋(join‘𝐾) 0 ) = 𝑋)
112, 10eqtr2d 2857 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1211adantrl 714 . . 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 36373 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋)))))
1716imp 409 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1817adantrr 715 . . 3 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1912, 18eqtr4d 2859 . 2 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑋 = 𝑌)
2019ex 415 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110   class class class wbr 5058  cfv 6349  (class class class)co 7150  Basecbs 16477  lecple 16566  occoc 16567  joincjn 17548  meetcmee 17549  0.cp0 17641  OLcol 36304  OMLcoml 36305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17532  df-poset 17550  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-lat 17650  df-oposet 36306  df-ol 36308  df-oml 36309
This theorem is referenced by:  omlfh1N  36388  atlatmstc  36449
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