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Theorem 1cvratlt 34270
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 999 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl3 1001 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 755 . . 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 34269 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. q  e.  A  q  .<  X )
101, 2, 3, 9syl3anc 1228 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  E. q  e.  A  q  .<  X )
11 simp1l1 1089 . . . 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 1090 . . . 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 997 . . . 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 1091 . . . 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 1062 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<_  X )
16 simp3 998 . . . 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 34234 . . . 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 1246 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<  X )
2019rexlimdv3a 2957 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   ` cfv 5586   Basecbs 14483   lecple 14555   ltcplt 15421   1.cp1 15518    <o ccvr 34059   Atomscatm 34060   HLchlt 34147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148
This theorem is referenced by:  cdlemb  34590  lhplt  34796
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