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Theorem cvrcmp2 30082
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 29980 . . . 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 2436 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
75, 6opoccl 29992 . . . 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 29992 . . . 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 29992 . . . 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 30075 . . . . . . 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 30075 . . . . . . 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 30081 . . 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 29998 . . 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 29994 . . 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 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   occoc 13537   Posetcpo 14397   OPcops 29970    <o ccvr 30060
This theorem is referenced by:  llncvrlpln  30355  lplncvrlvol  30413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-poset 14403  df-plt 14415  df-oposet 29974  df-covers 30064
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