Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1cvratlt Unicode version

Theorem 1cvratlt 29663
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 958 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl3 960 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 732 . . 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 29662 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. q  e.  A  q  .<  X )
101, 2, 3, 9syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1. 
/\  P  .<_  X ) )  ->  E. q  e.  A  q  .<  X )
11 simp1l1 1048 . . . 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 1049 . . . 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 956 . . . 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 1050 . . . 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 1021 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<_  X )
16 simp3 957 . . . 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 29627 . . . 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 1200 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) )  /\  q  e.  A  /\  q  .<  X )  ->  P  .<  X )
2019rexlimdv3a 2669 . 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 14 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   ltcplt 14075   1.cp1 14144    <o ccvr 29452   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  cdlemb  29983  lhplt  30189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
  Copyright terms: Public domain W3C validator