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Theorem omllaw3 35054
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 28635 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 6804 . . . . . 6 ((𝑌 ( 𝑋)) = 0 → (𝑋(join‘𝐾)(𝑌 ( 𝑋))) = (𝑋(join‘𝐾) 0 ))
21adantl 467 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → (𝑋(join‘𝐾)(𝑌 ( 𝑋))) = (𝑋(join‘𝐾) 0 ))
3 omlol 35049 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OL)
4 omllaw3.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
5 eqid 2771 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
6 omllaw3.z . . . . . . . . 9 0 = (0.‘𝐾)
74, 5, 6olj01 35034 . . . . . . . 8 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
83, 7sylan 569 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
983adant3 1126 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
109adantr 466 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → (𝑋(join‘𝐾) 0 ) = 𝑋)
112, 10eqtr2d 2806 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1211adantrl 695 . . 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 35052 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋)))))
1716imp 393 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1817adantrr 696 . . 3 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ( 𝑋))))
1912, 18eqtr4d 2808 . 2 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 )) → 𝑋 = 𝑌)
2019ex 397 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  occoc 16157  joincjn 17152  meetcmee 17153  0.cp0 17245  OLcol 34983  OMLcoml 34984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-preset 17136  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-oposet 34985  df-ol 34987  df-oml 34988
This theorem is referenced by:  omlfh1N  35067  atlatmstc  35128
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