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Theorem omllaw5N 29506
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 22306 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 955 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OML )
2 simp2 956 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 omllat 29501 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
4 omllaw5.b . . . . 5  |-  B  =  ( Base `  K
)
5 omllaw5.j . . . . 5  |-  .\/  =  ( join `  K )
64, 5latjcl 14255 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
73, 6syl3an1 1215 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
81, 2, 73jca 1132 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( K  e.  OML  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B ) )
9 eqid 2358 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
104, 9, 5latlej1 14265 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
113, 10syl3an1 1215 . 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 29503 . 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 56 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 4    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   lecple 13312   occoc 13313   joincjn 14177   meetcmee 14178   Latclat 14250   OMLcoml 29434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-lub 14207  df-join 14209  df-meet 14210  df-lat 14251  df-oposet 29435  df-ol 29437  df-oml 29438
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