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Theorem lecmtN 30116
Description: Ordered elements commute. (lecmi 23106 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
lecmt.b  |-  B  =  ( Base `  K
)
lecmt.l  |-  .<_  =  ( le `  K )
lecmt.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
lecmtN  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  X C Y ) )

Proof of Theorem lecmtN
StepHypRef Expression
1 omllat 30102 . . . . 5  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 979 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 simp2 959 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
4 omlop 30101 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  OP )
543ad2ant1 979 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
6 lecmt.b . . . . . . 7  |-  B  =  ( Base `  K
)
7 eqid 2438 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
86, 7opoccl 30054 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
95, 3, 8syl2anc 644 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
10 simp3 960 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 eqid 2438 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
126, 11latjcl 14481 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
132, 9, 10, 12syl3anc 1185 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
14 lecmt.l . . . . 5  |-  .<_  =  ( le `  K )
15 eqid 2438 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
166, 14, 15latmle1 14507 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X )
172, 3, 13, 16syl3anc 1185 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X )
186, 15latmcl 14482 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  e.  B )
192, 3, 13, 18syl3anc 1185 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  e.  B )
206, 14lattr 14487 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  e.  B  /\  X  e.  B  /\  Y  e.  B )
)  ->  ( (
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X  /\  X  .<_  Y )  ->  ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  .<_  Y )
)
212, 19, 3, 10, 20syl13anc 1187 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( X ( meet `  K
) ( ( ( oc `  K ) `
 X ) (
join `  K ) Y ) )  .<_  X  /\  X  .<_  Y )  ->  ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  .<_  Y )
)
2217, 21mpand 658 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  Y ) )
23 lecmt.c . . 3  |-  C  =  ( cm `  K
)
246, 14, 11, 15, 7, 23cmtbr4N 30115 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  Y ) )
2522, 24sylibrd 227 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  X C Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   occoc 13539   joincjn 14403   meetcmee 14404   Latclat 14476   OPcops 30032   cmccmtN 30033   OMLcoml 30035
This theorem is referenced by:  cmtidN  30117  omlmod1i2N  30120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-lat 14477  df-oposet 30036  df-cmtN 30037  df-ol 30038  df-oml 30039
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