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Theorem cvrcmp2 29767
Description: If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
Hypotheses
Ref Expression
cvrcmp.b  |-  B  =  ( Base `  K
)
cvrcmp.l  |-  .<_  =  ( le `  K )
cvrcmp.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrcmp2  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( X  .<_  Y  <->  X  =  Y ) )

Proof of Theorem cvrcmp2
StepHypRef Expression
1 opposet 29665 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
213ad2ant1 978 . . 3  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  K  e.  Poset )
3 simp1 957 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  K  e.  OP )
4 simp22 991 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  Y  e.  B )
5 cvrcmp.b . . . . 5  |-  B  =  ( Base `  K
)
6 eqid 2404 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
75, 6opoccl 29677 . . . 4  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
83, 4, 7syl2anc 643 . . 3  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( ( oc `  K ) `  Y
)  e.  B )
9 simp21 990 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  X  e.  B )
105, 6opoccl 29677 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
113, 9, 10syl2anc 643 . . 3  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( ( oc `  K ) `  X
)  e.  B )
12 simp23 992 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  Z  e.  B )
135, 6opoccl 29677 . . . 4  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
143, 12, 13syl2anc 643 . . 3  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( ( oc `  K ) `  Z
)  e.  B )
15 cvrcmp.c . . . . . . . 8  |-  C  =  (  <o  `  K )
165, 6, 15cvrcon3b 29760 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
( ( oc `  K ) `  Z
) C ( ( oc `  K ) `
 X ) ) )
17163adant3r2 1163 . . . . . 6  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Z  <->  ( ( oc `  K ) `  Z ) C ( ( oc `  K
) `  X )
) )
185, 6, 15cvrcon3b 29760 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  <-> 
( ( oc `  K ) `  Z
) C ( ( oc `  K ) `
 Y ) ) )
19183adant3r1 1162 . . . . . 6  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y C Z  <->  ( ( oc `  K ) `  Z ) C ( ( oc `  K
) `  Y )
) )
2017, 19anbi12d 692 . . . . 5  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X C Z  /\  Y C Z )  <->  ( ( ( oc `  K ) `
 Z ) C ( ( oc `  K ) `  X
)  /\  ( ( oc `  K ) `  Z ) C ( ( oc `  K
) `  Y )
) ) )
2120biimp3a 1283 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( ( ( oc
`  K ) `  Z ) C ( ( oc `  K
) `  X )  /\  ( ( oc `  K ) `  Z
) C ( ( oc `  K ) `
 Y ) ) )
2221ancomd 439 . . 3  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( ( ( oc
`  K ) `  Z ) C ( ( oc `  K
) `  Y )  /\  ( ( oc `  K ) `  Z
) C ( ( oc `  K ) `
 X ) ) )
23 cvrcmp.l . . . 4  |-  .<_  =  ( le `  K )
245, 23, 15cvrcmp 29766 . . 3  |-  ( ( K  e.  Poset  /\  (
( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Z
)  e.  B )  /\  ( ( ( oc `  K ) `
 Z ) C ( ( oc `  K ) `  Y
)  /\  ( ( oc `  K ) `  Z ) C ( ( oc `  K
) `  X )
) )  ->  (
( ( oc `  K ) `  Y
)  .<_  ( ( oc
`  K ) `  X )  <->  ( ( oc `  K ) `  Y )  =  ( ( oc `  K
) `  X )
) )
252, 8, 11, 14, 22, 24syl131anc 1197 . 2  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( ( ( oc
`  K ) `  Y )  .<_  ( ( oc `  K ) `
 X )  <->  ( ( oc `  K ) `  Y )  =  ( ( oc `  K
) `  X )
) )
265, 23, 6oplecon3b 29683 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( ( oc `  K ) `  Y )  .<_  ( ( oc `  K ) `
 X ) ) )
273, 9, 4, 26syl3anc 1184 . 2  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( X  .<_  Y  <->  ( ( oc `  K ) `  Y )  .<_  ( ( oc `  K ) `
 X ) ) )
285, 6opcon3b 29679 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  <-> 
( ( oc `  K ) `  Y
)  =  ( ( oc `  K ) `
 X ) ) )
293, 9, 4, 28syl3anc 1184 . 2  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( X  =  Y  <-> 
( ( oc `  K ) `  Y
)  =  ( ( oc `  K ) `
 X ) ) )
3025, 27, 293bitr4d 277 1  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  -> 
( X  .<_  Y  <->  X  =  Y ) )
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   Basecbs 13424   lecple 13491   occoc 13492   Posetcpo 14352   OPcops 29655    <o ccvr 29745
This theorem is referenced by:  llncvrlpln  30040  lplncvrlvol  30098
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-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-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  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-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-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-poset 14358  df-plt 14370  df-oposet 29659  df-covers 29749
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