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Theorem 1cvratlt 32748
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 1008 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl3 1010 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 762 . . 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 32747 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. q  e.  A  q  .<  X )
101, 2, 3, 9syl3anc 1264 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  E. q  e.  A  q  .<  X )
11 simp1l1 1098 . . . 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 1099 . . . 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 1006 . . . 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 1100 . . . 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 1071 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<_  X )
16 simp3 1007 . . . 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 32712 . . . 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 1282 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<  X )
2019rexlimdv3a 2917 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867   E.wrex 2774   class class class wbr 4417   ` cfv 5592   Basecbs 15073   lecple 15149   ltcplt 16130   1.cp1 16228    <o ccvr 32537   Atomscatm 32538   HLchlt 32625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-p1 16230  df-lat 16236  df-clat 16298  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626
This theorem is referenced by:  cdlemb  33068  lhplt  33274
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