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Theorem ncvr1 29755
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 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 ncvr1.u . . . 4  |-  .1.  =  ( 1. `  K )
41, 2, 3ople1 29674 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X ( le `  K )  .1.  )
5 opposet 29665 . . . . . 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 29668 . . . . . 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 2404 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
121, 2, 11pltnle 14378 . . . . 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 29753 . . 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 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   ltcplt 14353   1.cp1 14422   OPcops 29655    <o ccvr 29745
This theorem is referenced by:  lhp2lt  30483
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-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-p1 14424  df-oposet 29659  df-covers 29749
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