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Theorem cmt2N 28690
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 22133 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 28682 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 981 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 cmt2.b . . . . . . 7  |-  B  =  ( Base `  K
)
4 eqid 2258 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
53, 4latmcl 14120 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
61, 5syl3an1 1220 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
7 simp2 961 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 omlop 28681 . . . . . . . 8  |-  ( K  e.  OML  ->  K  e.  OP )
983ad2ant1 981 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
10 simp3 962 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 cmt2.o . . . . . . . 8  |-  ._|_  =  ( oc `  K )
123, 11opoccl 28634 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
139, 10, 12syl2anc 645 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
143, 4latmcl 14120 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X ( meet `  K
) (  ._|_  `  Y
) )  e.  B
)
152, 7, 13, 14syl3anc 1187 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  Y ) )  e.  B )
16 eqid 2258 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
173, 16latjcom 14128 . . . . 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 1187 . . . 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 28635 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
209, 10, 19syl2anc 645 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )
2120oveq2d 5808 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) (  ._|_  `  (  ._|_  `  Y ) ) )  =  ( X ( meet `  K
) Y ) )
2221oveq2d 5808 . . . 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 2293 . . 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 2269 . 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 28651 . 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 28651 . . 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 1232 . 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 278 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 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   occoc 13179   joincjn 14041   meetcmee 14042   Latclat 14114   OPcops 28612   cmccmtN 28613   OMLcoml 28615
This theorem is referenced by:  cmt3N  28691  cmt4N  28692  omlfh1N  28698
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-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-join 14073  df-lat 14115  df-oposet 28616  df-cmtN 28617  df-ol 28618  df-oml 28619
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