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Theorem omllaw4 34013
Description: Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
omllaw4.b 𝐵 = (Base‘𝐾)
omllaw4.l = (le‘𝐾)
omllaw4.m = (meet‘𝐾)
omllaw4.o = (oc‘𝐾)
Assertion
Ref Expression
omllaw4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋))

Proof of Theorem omllaw4
StepHypRef Expression
1 simp1 1059 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
2 omlop 34008 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ OP)
323ad2ant1 1080 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
4 simp3 1061 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
5 omllaw4.b . . . . 5 𝐵 = (Base‘𝐾)
6 omllaw4.o . . . . 5 = (oc‘𝐾)
75, 6opoccl 33961 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
83, 4, 7syl2anc 692 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
9 simp2 1060 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
105, 6opoccl 33961 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
113, 9, 10syl2anc 692 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
12 omllaw4.l . . . 4 = (le‘𝐾)
13 eqid 2621 . . . 4 (join‘𝐾) = (join‘𝐾)
14 omllaw4.m . . . 4 = (meet‘𝐾)
155, 12, 13, 14, 6omllaw 34010 . . 3 ((𝐾 ∈ OML ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
161, 8, 11, 15syl3anc 1323 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
175, 12, 6oplecon3b 33967 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
182, 17syl3an1 1356 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
19 omllat 34009 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
20193ad2ant1 1080 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
215, 14latmcl 16973 . . . . . . 7 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
2220, 11, 4, 21syl3anc 1323 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
235, 6opoccl 33961 . . . . . 6 ((𝐾 ∈ OP ∧ (( 𝑋) 𝑌) ∈ 𝐵) → ( ‘(( 𝑋) 𝑌)) ∈ 𝐵)
243, 22, 23syl2anc 692 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) ∈ 𝐵)
255, 14latmcl 16973 . . . . 5 ((𝐾 ∈ Lat ∧ ( ‘(( 𝑋) 𝑌)) ∈ 𝐵𝑌𝐵) → (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵)
2620, 24, 4, 25syl3anc 1323 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵)
275, 6opcon3b 33963 . . . 4 ((𝐾 ∈ OP ∧ (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵𝑋𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌))))
283, 26, 9, 27syl3anc 1323 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌))))
295, 13latjcom 16980 . . . . . 6 ((𝐾 ∈ Lat ∧ (( 𝑋) 𝑌) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
3020, 22, 8, 29syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
31 omlol 34007 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ OL)
32313ad2ant1 1080 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
335, 13, 14, 6oldmm2 33985 . . . . . 6 ((𝐾 ∈ OL ∧ (( 𝑋) 𝑌) ∈ 𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)))
3432, 22, 4, 33syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)))
355, 6opococ 33962 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
363, 4, 35syl2anc 692 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
3736oveq2d 6620 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ( ‘( 𝑌))) = (( 𝑋) 𝑌))
3837oveq2d 6620 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌)))) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
3930, 34, 383eqtr4d 2665 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌)))))
4039eqeq2d 2631 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) ↔ ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
4128, 40bitrd 268 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
4216, 18, 413imtr4d 283 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  occoc 15870  joincjn 16865  meetcmee 16866  Latclat 16966  OPcops 33939  OLcol 33941  OMLcoml 33942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-lat 16967  df-oposet 33943  df-ol 33945  df-oml 33946
This theorem is referenced by:  poml4N  34719  dihoml4c  36145
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