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Theorem cmtcomlemN 29743
Description: Lemma for cmtcomN 29744. (cmcmlem 23054 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtcom.b  |-  B  =  ( Base `  K
)
cmtcom.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
cmtcomlemN  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  Y C X ) )

Proof of Theorem cmtcomlemN
StepHypRef Expression
1 omllat 29737 . . . . . . . . . . . 12  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 978 . . . . . . . . . . 11  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 omlop 29736 . . . . . . . . . . . . 13  |-  ( K  e.  OML  ->  K  e.  OP )
4 cmtcom.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  K
)
5 eqid 2412 . . . . . . . . . . . . . 14  |-  ( oc
`  K )  =  ( oc `  K
)
64, 5opoccl 29689 . . . . . . . . . . . . 13  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
73, 6sylan 458 . . . . . . . . . . . 12  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
873adant3 977 . . . . . . . . . . 11  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
9 simp3 959 . . . . . . . . . . 11  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
10 eqid 2412 . . . . . . . . . . . 12  |-  ( le
`  K )  =  ( le `  K
)
11 eqid 2412 . . . . . . . . . . . 12  |-  ( join `  K )  =  (
join `  K )
124, 10, 11latlej2 14453 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  Y  e.  B )  ->  Y ( le `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )
132, 8, 9, 12syl3anc 1184 . . . . . . . . . 10  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y ( le `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )
144, 11latjcl 14442 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
152, 8, 9, 14syl3anc 1184 . . . . . . . . . . 11  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
16 eqid 2412 . . . . . . . . . . . 12  |-  ( meet `  K )  =  (
meet `  K )
174, 10, 16latleeqm2 14472 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )  -> 
( Y ( le
`  K ) ( ( ( oc `  K ) `  X
) ( join `  K
) Y )  <->  ( (
( ( oc `  K ) `  X
) ( join `  K
) Y ) (
meet `  K ) Y )  =  Y ) )
182, 9, 15, 17syl3anc 1184 . . . . . . . . . 10  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y ( le
`  K ) ( ( ( oc `  K ) `  X
) ( join `  K
) Y )  <->  ( (
( ( oc `  K ) `  X
) ( join `  K
) Y ) (
meet `  K ) Y )  =  Y ) )
1913, 18mpbid 202 . . . . . . . . 9  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( ( oc `  K ) `
 X ) (
join `  K ) Y ) ( meet `  K ) Y )  =  Y )
2019oveq2d 6064 . . . . . . . 8  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( ( oc `  K ) `  X
) ( join `  K
) Y ) (
meet `  K ) Y ) )  =  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) Y ) )
21 omlol 29735 . . . . . . . . . 10  |-  ( K  e.  OML  ->  K  e.  OL )
22213ad2ant1 978 . . . . . . . . 9  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OL )
2333ad2ant1 978 . . . . . . . . . . 11  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
244, 5opoccl 29689 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
2523, 9, 24syl2anc 643 . . . . . . . . . 10  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
264, 11latjcl 14442 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B )  ->  ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) )  e.  B
)
272, 8, 25, 26syl3anc 1184 . . . . . . . . 9  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) )  e.  B )
284, 16latmassOLD 29724 . . . . . . . . 9  |-  ( ( K  e.  OL  /\  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) )  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B  /\  Y  e.  B ) )  -> 
( ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ) (
meet `  K ) Y )  =  ( ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) ( meet `  K
) ( ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ( meet `  K ) Y ) ) )
2922, 27, 15, 9, 28syl13anc 1186 . . . . . . . 8  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ) (
meet `  K ) Y )  =  ( ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) ( meet `  K
) ( ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ( meet `  K ) Y ) ) )
304, 11, 16, 5oldmm1 29712 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X ( meet `  K
) Y ) )  =  ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) )
3121, 30syl3an1 1217 . . . . . . . . 9  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X ( meet `  K
) Y ) )  =  ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) )
3231oveq1d 6063 . . . . . . . 8  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  ( X ( meet `  K
) Y ) ) ( meet `  K
) Y )  =  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) Y ) )
3320, 29, 323eqtr4rd 2455 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  ( X ( meet `  K
) Y ) ) ( meet `  K
) Y )  =  ( ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ) (
meet `  K ) Y ) )
3433adantr 452 . . . . . 6  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( ( oc `  K ) `
 ( X (
meet `  K ) Y ) ) (
meet `  K ) Y )  =  ( ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ) (
meet `  K ) Y ) )
354, 11, 16, 5oldmj4 29719 . . . . . . . . . . . . 13  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X (
meet `  K ) Y ) )
3621, 35syl3an1 1217 . . . . . . . . . . . 12  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X (
meet `  K ) Y ) )
374, 11, 16, 5oldmj2 29717 . . . . . . . . . . . . 13  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) Y ) )  =  ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) )
3821, 37syl3an1 1217 . . . . . . . . . . . 12  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) Y ) )  =  ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) )
3936, 38oveq12d 6066 . . . . . . . . . . 11  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) ) ( join `  K
) ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) ) )  =  ( ( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )
4039eqeq2d 2423 . . . . . . . . . 10  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) ) ( join `  K
) ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) ) )  <->  X  =  ( ( X (
meet `  K ) Y ) ( join `  K ) ( X ( meet `  K
) ( ( oc
`  K ) `  Y ) ) ) ) )
4140biimpar 472 . . . . . . . . 9  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  X  =  ( ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) ) ( join `  K
) ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) ) ) )
4241fveq2d 5699 . . . . . . . 8  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( oc
`  K ) `  X )  =  ( ( oc `  K
) `  ( (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  Y
) ) ) (
join `  K )
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) Y ) ) ) ) )
434, 11, 16, 5oldmj4 29719 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) )  e.  B  /\  (
( ( oc `  K ) `  X
) ( join `  K
) Y )  e.  B )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  Y
) ) ) (
join `  K )
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) Y ) ) ) )  =  ( ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) ( meet `  K
) ( ( ( oc `  K ) `
 X ) (
join `  K ) Y ) ) )
4422, 27, 15, 43syl3anc 1184 . . . . . . . . 9  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) ) ( join `  K
) ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) ) ) )  =  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ) )
4544adantr 452 . . . . . . . 8  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 Y ) ) ) ( join `  K
) ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) ) ) )  =  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) ) )
4642, 45eqtr2d 2445 . . . . . . 7  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  Y
) ) ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  =  ( ( oc `  K ) `  X
) )
4746oveq1d 6063 . . . . . 6  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( ( ( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  Y ) ) (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) ) ( meet `  K
) Y )  =  ( ( ( oc
`  K ) `  X ) ( meet `  K ) Y ) )
4834, 47eqtrd 2444 . . . . 5  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( ( oc `  K ) `
 ( X (
meet `  K ) Y ) ) (
meet `  K ) Y )  =  ( ( ( oc `  K ) `  X
) ( meet `  K
) Y ) )
4948oveq2d 6064 . . . 4  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( X ( meet `  K
) Y ) (
join `  K )
( ( ( oc
`  K ) `  ( X ( meet `  K
) Y ) ) ( meet `  K
) Y ) )  =  ( ( X ( meet `  K
) Y ) (
join `  K )
( ( ( oc
`  K ) `  X ) ( meet `  K ) Y ) ) )
50 simp1 957 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OML )
514, 16latmcl 14443 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
521, 51syl3an1 1217 . . . . . . 7  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  e.  B )
5350, 52, 93jca 1134 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( K  e.  OML  /\  ( X ( meet `  K ) Y )  e.  B  /\  Y  e.  B ) )
544, 10, 16latmle2 14469 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y ) ( le `  K
) Y )
551, 54syl3an1 1217 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y ) ( le `  K
) Y )
564, 10, 11, 16, 5omllaw2N 29739 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X ( meet `  K
) Y )  e.  B  /\  Y  e.  B )  ->  (
( X ( meet `  K ) Y ) ( le `  K
) Y  ->  (
( X ( meet `  K ) Y ) ( join `  K
) ( ( ( oc `  K ) `
 ( X (
meet `  K ) Y ) ) (
meet `  K ) Y ) )  =  Y ) )
5753, 55, 56sylc 58 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K ) Y ) ( join `  K ) ( ( ( oc `  K
) `  ( X
( meet `  K ) Y ) ) (
meet `  K ) Y ) )  =  Y )
5857adantr 452 . . . 4  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( X ( meet `  K
) Y ) (
join `  K )
( ( ( oc
`  K ) `  ( X ( meet `  K
) Y ) ) ( meet `  K
) Y ) )  =  Y )
594, 16latmcom 14467 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  =  ( Y (
meet `  K ) X ) )
601, 59syl3an1 1217 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) Y )  =  ( Y (
meet `  K ) X ) )
614, 16latmcom 14467 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( meet `  K ) Y )  =  ( Y (
meet `  K )
( ( oc `  K ) `  X
) ) )
622, 8, 9, 61syl3anc 1184 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( meet `  K ) Y )  =  ( Y (
meet `  K )
( ( oc `  K ) `  X
) ) )
6360, 62oveq12d 6066 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X (
meet `  K ) Y ) ( join `  K ) ( ( ( oc `  K
) `  X )
( meet `  K ) Y ) )  =  ( ( Y (
meet `  K ) X ) ( join `  K ) ( Y ( meet `  K
) ( ( oc
`  K ) `  X ) ) ) )
6463adantr 452 . . . 4  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  ( ( X ( meet `  K
) Y ) (
join `  K )
( ( ( oc
`  K ) `  X ) ( meet `  K ) Y ) )  =  ( ( Y ( meet `  K
) X ) (
join `  K )
( Y ( meet `  K ) ( ( oc `  K ) `
 X ) ) ) )
6549, 58, 643eqtr3d 2452 . . 3  |-  ( ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  /\  X  =  (
( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  ->  Y  =  ( ( Y ( meet `  K ) X ) ( join `  K
) ( Y (
meet `  K )
( ( oc `  K ) `  X
) ) ) )
6665ex 424 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( ( X ( meet `  K ) Y ) ( join `  K
) ( X (
meet `  K )
( ( oc `  K ) `  Y
) ) )  ->  Y  =  ( ( Y ( meet `  K
) X ) (
join `  K )
( Y ( meet `  K ) ( ( oc `  K ) `
 X ) ) ) ) )
67 cmtcom.c . . 3  |-  C  =  ( cm `  K
)
684, 11, 16, 5, 67cmtvalN 29706 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
X  =  ( ( X ( meet `  K
) Y ) (
join `  K )
( X ( meet `  K ) ( ( oc `  K ) `
 Y ) ) ) ) )
694, 11, 16, 5, 67cmtvalN 29706 . . 3  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y C X  <-> 
Y  =  ( ( Y ( meet `  K
) X ) (
join `  K )
( Y ( meet `  K ) ( ( oc `  K ) `
 X ) ) ) ) )
70693com23 1159 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y C X  <-> 
Y  =  ( ( Y ( meet `  K
) X ) (
join `  K )
( Y ( meet `  K ) ( ( oc `  K ) `
 X ) ) ) ) )
7166, 68, 703imtr4d 260 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  Y C X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   occoc 13500   joincjn 14364   meetcmee 14365   Latclat 14437   OPcops 29667   cmccmtN 29668   OLcol 29669   OMLcoml 29670
This theorem is referenced by:  cmtcomN  29744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-lat 14438  df-oposet 29671  df-cmtN 29672  df-ol 29673  df-oml 29674
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