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Theorem cmt2N 29513
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 22174 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmt2.b  |-  B  =  ( Base `  K
)
cmt2.o  |-  ._|_  =  ( oc `  K )
cmt2.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
cmt2N  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X C (  ._|_  `  Y ) ) )

Proof of Theorem cmt2N
StepHypRef Expression
1 omllat 29505 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 976 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 cmt2.b . . . . . . 7  |-  B  =  ( Base `  K
)
4 eqid 2285 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
53, 4latmcl 14159 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
61, 5syl3an1 1215 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
7 simp2 956 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 omlop 29504 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
983ad2ant1 976 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
10 simp3 957 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 cmt2.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
123, 11opoccl 29457 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
139, 10, 12syl2anc 642 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
143, 4latmcl 14159 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X ( meet `  K
) (  ._|_  `  Y
) )  e.  B
)
152, 7, 13, 14syl3anc 1182 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  Y ) )  e.  B )
16 eqid 2285 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
173, 16latjcom 14167 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X ( meet `  K
) Y )  e.  B  /\  ( X ( meet `  K
) (  ._|_  `  Y
) )  e.  B
)  ->  ( ( X ( meet `  K
) Y ) (
join `  K )
( X ( meet `  K ) (  ._|_  `  Y ) ) )  =  ( ( X ( meet `  K
) (  ._|_  `  Y
) ) ( join `  K ) ( X ( meet `  K
) Y ) ) )
182, 6, 15, 17syl3anc 1182 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K ) Y ) ( join `  K ) ( X ( meet `  K
) (  ._|_  `  Y
) ) )  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K ) Y ) ) )
193, 11opococ 29458 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
209, 10, 19syl2anc 642 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
2120oveq2d 5876 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  (  ._|_  `  Y ) ) )  =  ( X ( meet `  K
) Y ) )
2221oveq2d 5876 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) )  =  ( ( X ( meet `  K
) (  ._|_  `  Y
) ) ( join `  K ) ( X ( meet `  K
) Y ) ) )
2318, 22eqtr4d 2320 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K ) Y ) ( join `  K ) ( X ( meet `  K
) (  ._|_  `  Y
) ) )  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) ) )
2423eqeq2d 2296 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( ( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  Y )
) )  <->  X  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) ) ) )
25 cmt2.c . . 3  |-  C  =  ( cm `  K
)
263, 16, 4, 11, 25cmtvalN 29474 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X  =  ( ( X ( meet `  K
) Y ) (
join `  K )
( X ( meet `  K ) (  ._|_  `  Y ) ) ) ) )
273, 16, 4, 11, 25cmtvalN 29474 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X C (  ._|_  `  Y
)  <->  X  =  (
( X ( meet `  K ) (  ._|_  `  Y ) ) (
join `  K )
( X ( meet `  K ) (  ._|_  `  (  ._|_  `  Y ) ) ) ) ) )
2813, 27syld3an3 1227 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( 
._|_  `  Y )  <->  X  =  ( ( X (
meet `  K )
(  ._|_  `  Y )
) ( join `  K
) ( X (
meet `  K )
(  ._|_  `  (  ._|_  `  Y ) ) ) ) ) )
2924, 26, 283bitr4d 276 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X C (  ._|_  `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1625    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Basecbs 13150   occoc 13218   joincjn 14080   meetcmee 14081   Latclat 14153   OPcops 29435   cmccmtN 29436   OMLcoml 29438
This theorem is referenced by:  cmt3N  29514  cmt4N  29515  omlfh1N  29521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-join 14112  df-lat 14154  df-oposet 29439  df-cmtN 29440  df-ol 29441  df-oml 29442
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