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Theorem omllaw4 36250
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 1130 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
2 omlop 36245 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ OP)
323ad2ant1 1127 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
4 simp3 1132 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
5 omllaw4.b . . . . 5 𝐵 = (Base‘𝐾)
6 omllaw4.o . . . . 5 = (oc‘𝐾)
75, 6opoccl 36198 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
83, 4, 7syl2anc 584 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
9 simp2 1131 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
105, 6opoccl 36198 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
113, 9, 10syl2anc 584 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
12 omllaw4.l . . . 4 = (le‘𝐾)
13 eqid 2824 . . . 4 (join‘𝐾) = (join‘𝐾)
14 omllaw4.m . . . 4 = (meet‘𝐾)
155, 12, 13, 14, 6omllaw 36247 . . 3 ((𝐾 ∈ OML ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
161, 8, 11, 15syl3anc 1365 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
175, 12, 6oplecon3b 36204 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
182, 17syl3an1 1157 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
19 omllat 36246 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
20193ad2ant1 1127 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
215, 14latmcl 17654 . . . . . . 7 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
2220, 11, 4, 21syl3anc 1365 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
235, 6opoccl 36198 . . . . . 6 ((𝐾 ∈ OP ∧ (( 𝑋) 𝑌) ∈ 𝐵) → ( ‘(( 𝑋) 𝑌)) ∈ 𝐵)
243, 22, 23syl2anc 584 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) ∈ 𝐵)
255, 14latmcl 17654 . . . . 5 ((𝐾 ∈ Lat ∧ ( ‘(( 𝑋) 𝑌)) ∈ 𝐵𝑌𝐵) → (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵)
2620, 24, 4, 25syl3anc 1365 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵)
275, 6opcon3b 36200 . . . 4 ((𝐾 ∈ OP ∧ (( ‘(( 𝑋) 𝑌)) 𝑌) ∈ 𝐵𝑋𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌))))
283, 26, 9, 27syl3anc 1365 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌))))
295, 13latjcom 17661 . . . . . 6 ((𝐾 ∈ Lat ∧ (( 𝑋) 𝑌) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
3020, 22, 8, 29syl3anc 1365 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
31 omlol 36244 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ OL)
32313ad2ant1 1127 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
335, 13, 14, 6oldmm2 36222 . . . . . 6 ((𝐾 ∈ OL ∧ (( 𝑋) 𝑌) ∈ 𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)))
3432, 22, 4, 33syl3anc 1365 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = ((( 𝑋) 𝑌)(join‘𝐾)( 𝑌)))
355, 6opococ 36199 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
363, 4, 35syl2anc 584 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
3736oveq2d 7167 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ( ‘( 𝑌))) = (( 𝑋) 𝑌))
3837oveq2d 7167 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌)))) = (( 𝑌)(join‘𝐾)(( 𝑋) 𝑌)))
3930, 34, 383eqtr4d 2870 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌)))))
4039eqeq2d 2835 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = ( ‘(( ‘(( 𝑋) 𝑌)) 𝑌)) ↔ ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
4128, 40bitrd 280 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋 ↔ ( 𝑋) = (( 𝑌)(join‘𝐾)(( 𝑋) ( ‘( 𝑌))))))
4216, 18, 413imtr4d 295 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1081   = wceq 1530  wcel 2106   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  occoc 16565  joincjn 17546  meetcmee 17547  Latclat 17647  OPcops 36176  OLcol 36178  OMLcoml 36179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-proset 17530  df-poset 17548  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-lat 17648  df-oposet 36180  df-ol 36182  df-oml 36183
This theorem is referenced by:  poml4N  36957  dihoml4c  38380
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