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Theorem ncvr1 29388
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 2388 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 ncvr1.u . . . 4  |-  .1.  =  ( 1. `  K )
41, 2, 3ople1 29307 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X ( le `  K )  .1.  )
5 opposet 29298 . . . . . 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 29301 . . . . . 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 2388 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
121, 2, 11pltnle 14351 . . . . 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 29386 . . 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 1717   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464   Posetcpo 14325   ltcplt 14326   1.cp1 14395   OPcops 29288    <o ccvr 29378
This theorem is referenced by:  lhp2lt  30116
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-p1 14397  df-oposet 29292  df-covers 29382
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