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Theorem ncvr1 32838
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 2451 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 ncvr1.u . . . 4  |-  .1.  =  ( 1. `  K )
41, 2, 3ople1 32757 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X ( le `  K )  .1.  )
5 opposet 32747 . . . . . 6  |-  ( K  e.  OP  ->  K  e.  Poset )
65ad2antrr 732 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  K  e.  Poset
)
71, 3op1cl 32751 . . . . . 6  |-  ( K  e.  OP  ->  .1.  e.  B )
87ad2antrr 732 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  .1.  e.  B )
9 simplr 762 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  X  e.  B )
10 simpr 463 . . . . 5  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  .1.  ( lt `  K ) X )
11 eqid 2451 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
121, 2, 11pltnle 16212 . . . . 5  |-  ( ( ( K  e.  Poset  /\  .1.  e.  B  /\  X  e.  B )  /\  .1.  ( lt `  K ) X )  ->  -.  X ( le `  K )  .1.  )
136, 8, 9, 10, 12syl31anc 1271 . . . 4  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  ( lt
`  K ) X )  ->  -.  X
( le `  K
)  .1.  )
1413ex 436 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  ( lt
`  K ) X  ->  -.  X ( le `  K )  .1.  ) )
154, 14mt2d 121 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  ( lt
`  K ) X )
16 simpll 760 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  K  e.  OP )
177ad2antrr 732 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  e.  B )
18 simplr 762 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  X  e.  B )
19 simpr 463 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  C X )
20 ncvr1.c . . . 4  |-  C  =  (  <o  `  K )
211, 11, 20cvrlt 32836 . . 3  |-  ( ( ( K  e.  OP  /\  .1.  e.  B  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  ( lt `  K ) X )
2216, 17, 18, 19, 21syl31anc 1271 . 2  |-  ( ( ( K  e.  OP  /\  X  e.  B )  /\  .1.  C X )  ->  .1.  ( lt `  K ) X )
2315, 22mtand 665 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  C X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   class class class wbr 4402   ` cfv 5582   Basecbs 15121   lecple 15197   Posetcpo 16185   ltcplt 16186   1.cp1 16284   OPcops 32738    <o ccvr 32828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-p1 16286  df-oposet 32742  df-covers 32832
This theorem is referenced by:  lhp2lt  33566
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