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Theorem 1cvratlt 33457
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 991 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl3 993 . . 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 33456 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. q  e.  A  q  .<  X )
101, 2, 3, 9syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  E. q  e.  A  q  .<  X )
11 simp1l1 1081 . . . 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 1082 . . . 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 989 . . . 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 1083 . . . 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 1054 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<_  X )
16 simp3 990 . . . 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 33421 . . . 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 1237 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<  X )
2019rexlimdv3a 2949 . 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 965    = wceq 1370    e. wcel 1758   E.wrex 2800   class class class wbr 4401   ` cfv 5527   Basecbs 14293   lecple 14365   ltcplt 15231   1.cp1 15328    <o ccvr 33246   Atomscatm 33247   HLchlt 33334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335
This theorem is referenced by:  cdlemb  33777  lhplt  33983
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