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Theorem ncvr1 29971
Description: No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)
Hypotheses
Ref Expression
ncvr1.b  |-  B  =  ( Base `  K
)
ncvr1.u  |-  .1.  =  ( 1. `  K )
ncvr1.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
ncvr1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  C X )

Proof of Theorem ncvr1
StepHypRef Expression
1 ncvr1.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2435 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 ncvr1.u . . . 4  |-  .1.  =  ( 1. `  K )
41, 2, 3ople1 29890 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X ( le `  K )  .1.  )
5 opposet 29881 . . . . . 6  |-  ( K  e.  OP  ->  K  e.  Poset )
65ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  K  e.  Poset
)
71, 3op1cl 29884 . . . . . 6  |-  ( K  e.  OP  ->  .1.  e.  B )
87ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  .1.  e.  B )
9 simplr 732 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  X  e.  B )
10 simpr 448 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  .1.  ( lt `  K ) X )
11 eqid 2435 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
121, 2, 11pltnle 14413 . . . . 5  |-  ( ( ( K  e.  Poset  /\  .1.  e.  B  /\  X  e.  B )  /\  .1.  ( lt `  K ) X )  ->  -.  X ( le `  K )  .1.  )
136, 8, 9, 10, 12syl31anc 1187 . . . 4  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  -.  X
( le `  K
)  .1.  )
1413ex 424 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  ( lt
`  K ) X  ->  -.  X ( le `  K )  .1.  ) )
154, 14mt2d 111 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  ( lt
`  K ) X )
16 simpll 731 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  K  e.  OP )
177ad2antrr 707 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  e.  B )
18 simplr 732 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  X  e.  B )
19 simpr 448 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  C X )
20 ncvr1.c . . . 4  |-  C  =  (  <o  `  K )
211, 11, 20cvrlt 29969 . . 3  |-  ( ( ( K  e.  OP  /\  .1.  e.  B  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  ( lt `  K ) X )
2216, 17, 18, 19, 21syl31anc 1187 . 2  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  ( lt `  K ) X )
2315, 22mtand 641 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  C X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13459   lecple 13526   Posetcpo 14387   ltcplt 14388   1.cp1 14457   OPcops 29871    <o ccvr 29961
This theorem is referenced by:  lhp2lt  30699
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-undef 6535  df-riota 6541  df-poset 14393  df-plt 14405  df-lub 14421  df-p1 14459  df-oposet 29875  df-covers 29965
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