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Theorem cvrcmp2 29474
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 29372 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
213ad2ant1 976 . . 3  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  K  e.  Poset )
3 simp1 955 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  K  e.  OP )
4 simp22 989 . . . 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 2283 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
75, 6opoccl 29384 . . . 4  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
83, 4, 7syl2anc 642 . . 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 988 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  X  e.  B )
105, 6opoccl 29384 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
113, 9, 10syl2anc 642 . . 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 990 . . . 4  |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X C Z  /\  Y C Z ) )  ->  Z  e.  B )
135, 6opoccl 29384 . . . 4  |-  ( ( K  e.  OP  /\  Z  e.  B )  ->  ( ( oc `  K ) `  Z
)  e.  B )
143, 12, 13syl2anc 642 . . 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 29467 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
( ( oc `  K ) `  Z
) C ( ( oc `  K ) `
 X ) ) )
17163adant3r2 1161 . . . . . 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 29467 . . . . . . 7  |-  ( ( K  e.  OP  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  <-> 
( ( oc `  K ) `  Z
) C ( ( oc `  K ) `
 Y ) ) )
19183adant3r1 1160 . . . . . 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 691 . . . . 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 1281 . . . 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 438 . . 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 29473 . . 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 1195 . 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 29390 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( ( oc `  K ) `  Y )  .<_  ( ( oc `  K ) `
 X ) ) )
273, 9, 4, 26syl3anc 1182 . 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 29386 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  <-> 
( ( oc `  K ) `  Y
)  =  ( ( oc `  K ) `
 X ) ) )
293, 9, 4, 28syl3anc 1182 . 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 276 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   occoc 13216   Posetcpo 14074   OPcops 29362    <o ccvr 29452
This theorem is referenced by:  llncvrlpln  29747  lplncvrlvol  29805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-oposet 29366  df-covers 29456
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