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Theorem omllaw4 33551
 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 1054 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
2 omlop 33546 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ OP)
323ad2ant1 1075 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
4 simp3 1056 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
5 omllaw4.b . . . . 5 𝐵 = (Base‘𝐾)
6 omllaw4.o . . . . 5 = (oc‘𝐾)
75, 6opoccl 33499 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
83, 4, 7syl2anc 691 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
9 simp2 1055 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
105, 6opoccl 33499 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
113, 9, 10syl2anc 691 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
12 omllaw4.l . . . 4 = (le‘𝐾)
13 eqid 2610 . . . 4 (join‘𝐾) = (join‘𝐾)
14 omllaw4.m . . . 4 = (meet‘𝐾)
155, 12, 13, 14, 6omllaw 33548 . . 3 ((𝐾 ∈ OML ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
161, 8, 11, 15syl3anc 1318 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
175, 12, 6oplecon3b 33505 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
182, 17syl3an1 1351 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
19 omllat 33547 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
20193ad2ant1 1075 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
215, 14latmcl 16875 . . . . . . 7 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
2220, 11, 4, 21syl3anc 1318 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
235, 6opoccl 33499 . . . . . 6 ((𝐾 ∈ OP ∧ (( 𝑋) 𝑌) ∈ 𝐵) → ( ‘(( 𝑋) 𝑌)) ∈ 𝐵)
243, 22, 23syl2anc 691 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) ∈ 𝐵)
255, 14latmcl 16875 . . . . 5 ((𝐾 ∈ Lat ∧ ( ‘(( 𝑋) 𝑌)) ∈ 𝐵𝑌𝐵) → (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵)
2620, 24, 4, 25syl3anc 1318 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵)
275, 6opcon3b 33501 . . . 4 ((𝐾 ∈ OP ∧ (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵𝑋𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌))))
283, 26, 9, 27syl3anc 1318 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌))))
295, 13latjcom 16882 . . . . . 6 ((𝐾 ∈ Lat ∧ (( 𝑋) 𝑌) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
3020, 22, 8, 29syl3anc 1318 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
31 omlol 33545 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ OL)
32313ad2ant1 1075 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
335, 13, 14, 6oldmm2 33523 . . . . . 6 ((𝐾 ∈ OL ∧ (( 𝑋) 𝑌) ∈ 𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)))
3432, 22, 4, 33syl3anc 1318 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)))
355, 6opococ 33500 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
363, 4, 35syl2anc 691 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
3736oveq2d 6565 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ( ‘( 𝑌))) = (( 𝑋) 𝑌))
3837oveq2d 6565 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌)))) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
3930, 34, 383eqtr4d 2654 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌)))))
4039eqeq2d 2620 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) ↔ ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
4128, 40bitrd 267 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
4216, 18, 413imtr4d 282 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ 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  Latclat 16868  OPcops 33477  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-lat 16869  df-oposet 33481  df-ol 33483  df-oml 33484 This theorem is referenced by:  poml4N  34257  dihoml4c  35683
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