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Theorem cmtcomlemN 29731
Description: Lemma for cmtcomN 29732. (cmcmlem 23046 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 29725 . . . . . . . . . . . 12  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 978 . . . . . . . . . . 11  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 omlop 29724 . . . . . . . . . . . . 13  |-  ( K  e.  OML  ->  K  e.  OP )
4 cmtcom.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  K
)
5 eqid 2404 . . . . . . . . . . . . . 14  |-  ( oc
`  K )  =  ( oc `  K
)
64, 5opoccl 29677 . . . . . . . . . . . . 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 2404 . . . . . . . . . . . 12  |-  ( le
`  K )  =  ( le `  K
)
11 eqid 2404 . . . . . . . . . . . 12  |-  ( join `  K )  =  (
join `  K )
124, 10, 11latlej2 14445 . . . . . . . . . . 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 14434 . . . . . . . . . . . 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 2404 . . . . . . . . . . . 12  |-  ( meet `  K )  =  (
meet `  K )
174, 10, 16latleeqm2 14464 . . . . . . . . . . 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 6056 . . . . . . . 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 29723 . . . . . . . . . 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 29677 . . . . . . . . . . 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 14434 . . . . . . . . . 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 29712 . . . . . . . . 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 29700 . . . . . . . . . 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 6055 . . . . . . . 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 2447 . . . . . . 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 29707 . . . . . . . . . . . . 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 29705 . . . . . . . . . . . . 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 6058 . . . . . . . . . . 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 2415 . . . . . . . . . 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 5691 . . . . . . . 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 29707 . . . . . . . . . 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 2437 . . . . . . 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 6055 . . . . . 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 2436 . . . . 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 6056 . . . 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 14435 . . . . . . . 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 14461 . . . . . . 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 29727 . . . . . 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 14459 . . . . . . 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 14459 . . . . . . 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 6058 . . . . 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 2444 . . 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 29694 . 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 29694 . . 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 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   occoc 13492   joincjn 14356   meetcmee 14357   Latclat 14429   OPcops 29655   cmccmtN 29656   OLcol 29657   OMLcoml 29658
This theorem is referenced by:  cmtcomN  29732
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-lat 14430  df-oposet 29659  df-cmtN 29660  df-ol 29661  df-oml 29662
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