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Theorem 1cvratlt 30271
Description: An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)
Hypotheses
Ref Expression
1cvratlt.b  |-  B  =  ( Base `  K
)
1cvratlt.l  |-  .<_  =  ( le `  K )
1cvratlt.s  |-  .<  =  ( lt `  K )
1cvratlt.u  |-  .1.  =  ( 1. `  K )
1cvratlt.c  |-  C  =  (  <o  `  K )
1cvratlt.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
1cvratlt  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  P  .<  X )

Proof of Theorem 1cvratlt
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl3 962 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 733 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  X C  .1.  )
4 1cvratlt.b . . . 4  |-  B  =  ( Base `  K
)
5 1cvratlt.s . . . 4  |-  .<  =  ( lt `  K )
6 1cvratlt.u . . . 4  |-  .1.  =  ( 1. `  K )
7 1cvratlt.c . . . 4  |-  C  =  (  <o  `  K )
8 1cvratlt.a . . . 4  |-  A  =  ( Atoms `  K )
94, 5, 6, 7, 81cvratex 30270 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. q  e.  A  q  .<  X )
101, 2, 3, 9syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  E. q  e.  A  q  .<  X )
11 simp1l1 1050 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  K  e.  HL )
12 simp1l2 1051 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  e.  A )
13 simp2 958 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  -> 
q  e.  A )
14 simp1l3 1052 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  X  e.  B )
15 simp1rr 1023 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<_  X )
16 simp3 959 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  -> 
q  .<  X )
17 1cvratlt.l . . . . 5  |-  .<_  =  ( le `  K )
184, 17, 5, 8atlelt 30235 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  q  .<  X ) )  ->  P  .<  X )
1911, 12, 13, 14, 15, 16, 18syl132anc 1202 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<  X )
2019rexlimdv3a 2832 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  ( E. q  e.  A  q  .<  X  ->  P  .<  X ) )
2110, 20mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  P  .<  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   ltcplt 14398   1.cp1 14467    <o ccvr 30060   Atomscatm 30061   HLchlt 30148
This theorem is referenced by:  cdlemb  30591  lhplt  30797
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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