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Theorem ncvr1 30084
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 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 ncvr1.u . . . 4  |-  .1.  =  ( 1. `  K )
41, 2, 3ople1 30003 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X ( le `  K )  .1.  )
5 opposet 29994 . . . . . 6  |-  ( K  e.  OP  ->  K  e.  Poset )
65ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  K  e.  Poset
)
71, 3op1cl 29997 . . . . . 6  |-  ( K  e.  OP  ->  .1.  e.  B )
87ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  .1.  e.  B )
9 simplr 731 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  X  e.  B )
10 simpr 447 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  .1.  ( lt `  K ) X )
11 eqid 2296 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
121, 2, 11pltnle 14116 . . . . 5  |-  ( ( ( K  e.  Poset  /\  .1.  e.  B  /\  X  e.  B )  /\  .1.  ( lt `  K ) X )  ->  -.  X ( le `  K )  .1.  )
136, 8, 9, 10, 12syl31anc 1185 . . . 4  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  -.  X
( le `  K
)  .1.  )
1413ex 423 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  ( lt
`  K ) X  ->  -.  X ( le `  K )  .1.  ) )
154, 14mt2d 109 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  ( lt
`  K ) X )
16 simpll 730 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  K  e.  OP )
177ad2antrr 706 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  e.  B )
18 simplr 731 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  X  e.  B )
19 simpr 447 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  C X )
20 ncvr1.c . . . 4  |-  C  =  (  <o  `  K )
211, 11, 20cvrlt 30082 . . 3  |-  ( ( ( K  e.  OP  /\  .1.  e.  B  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  ( lt `  K ) X )
2216, 17, 18, 19, 21syl31anc 1185 . 2  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  ( lt `  K ) X )
2315, 22mtand 640 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  C X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   ltcplt 14091   1.cp1 14160   OPcops 29984    <o ccvr 30074
This theorem is referenced by:  lhp2lt  30812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-p1 14162  df-oposet 29988  df-covers 30078
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