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Theorem omllaw5N 28687
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 22153 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omllaw5.b  |-  B  =  ( Base `  K
)
omllaw5.j  |-  .\/  =  ( join `  K )
omllaw5.m  |-  ./\  =  ( meet `  K )
omllaw5.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
omllaw5N  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  ( X  .\/  Y
) ) )  =  ( X  .\/  Y
) )

Proof of Theorem omllaw5N
StepHypRef Expression
1 simp1 960 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OML )
2 simp2 961 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 omllat 28682 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
4 omllaw5.b . . . . 5  |-  B  =  ( Base `  K
)
5 omllaw5.j . . . . 5  |-  .\/  =  ( join `  K )
64, 5latjcl 14119 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
73, 6syl3an1 1220 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
81, 2, 73jca 1137 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( K  e.  OML  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B ) )
9 eqid 2258 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
104, 9, 5latlej1 14129 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
113, 10syl3an1 1220 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
12 omllaw5.m . . 3  |-  ./\  =  ( meet `  K )
13 omllaw5.o . . 3  |-  ._|_  =  ( oc `  K )
144, 9, 5, 12, 13omllaw2N 28684 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X ( le
`  K ) ( X  .\/  Y )  ->  ( X  .\/  ( (  ._|_  `  X
)  ./\  ( X  .\/  Y ) ) )  =  ( X  .\/  Y ) ) )
158, 11, 14sylc 58 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  ( X  .\/  Y
) ) )  =  ( X  .\/  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   occoc 13179   joincjn 14041   meetcmee 14042   Latclat 14114   OMLcoml 28615
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-lub 14071  df-join 14073  df-meet 14074  df-lat 14115  df-oposet 28616  df-ol 28618  df-oml 28619
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