Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hl2at Unicode version

Theorem hl2at 29646
Description: A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
Hypothesis
Ref Expression
hl2atom.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hl2at  |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  p  =/=  q )
Distinct variable groups:    q, p, A    K, p, q

Proof of Theorem hl2at
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2358 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2358 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 eqid 2358 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2358 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
51, 2, 3, 4hlhgt2 29630 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) ( ( 0.
`  K ) ( lt `  K ) x  /\  x ( lt `  K ) ( 1. `  K
) ) )
6 simpl 443 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  HL )
7 hlop 29604 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
87adantr 451 . . . . . . 7  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
91, 3op0cl 29426 . . . . . . 7  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
108, 9syl 15 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( 0. `  K
)  e.  ( Base `  K ) )
11 simpr 447 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  x  e.  ( Base `  K ) )
12 eqid 2358 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
13 hl2atom.a . . . . . . 7  |-  A  =  ( Atoms `  K )
141, 12, 2, 13hlrelat1 29641 . . . . . 6  |-  ( ( K  e.  HL  /\  ( 0. `  K )  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( ( 0. `  K ) ( lt `  K ) x  ->  E. p  e.  A  ( -.  p ( le `  K ) ( 0.
`  K )  /\  p ( le `  K ) x ) ) )
156, 10, 11, 14syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( ( 0. `  K ) ( lt
`  K ) x  ->  E. p  e.  A  ( -.  p ( le `  K ) ( 0. `  K )  /\  p ( le
`  K ) x ) ) )
161, 4op1cl 29427 . . . . . . 7  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
178, 16syl 15 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( 1. `  K
)  e.  ( Base `  K ) )
181, 12, 2, 13hlrelat1 29641 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K )  /\  ( 1. `  K )  e.  ( Base `  K
) )  ->  (
x ( lt `  K ) ( 1.
`  K )  ->  E. q  e.  A  ( -.  q ( le `  K ) x  /\  q ( le
`  K ) ( 1. `  K ) ) ) )
1917, 18mpd3an3 1278 . . . . 5  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( x ( lt
`  K ) ( 1. `  K )  ->  E. q  e.  A  ( -.  q ( le `  K ) x  /\  q ( le
`  K ) ( 1. `  K ) ) ) )
2015, 19anim12d 546 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( ( ( 0.
`  K ) ( lt `  K ) x  /\  x ( lt `  K ) ( 1. `  K
) )  ->  ( E. p  e.  A  ( -.  p ( le `  K ) ( 0. `  K )  /\  p ( le
`  K ) x )  /\  E. q  e.  A  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) ) ) )
21 reeanv 2783 . . . . 5  |-  ( E. p  e.  A  E. q  e.  A  (
( -.  p ( le `  K ) ( 0. `  K
)  /\  p ( le `  K ) x )  /\  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) )  <->  ( E. p  e.  A  ( -.  p ( le `  K ) ( 0.
`  K )  /\  p ( le `  K ) x )  /\  E. q  e.  A  ( -.  q
( le `  K
) x  /\  q
( le `  K
) ( 1. `  K ) ) ) )
22 nbrne2 4120 . . . . . . . 8  |-  ( ( p ( le `  K ) x  /\  -.  q ( le `  K ) x )  ->  p  =/=  q
)
2322ad2ant2lr 728 . . . . . . 7  |-  ( ( ( -.  p ( le `  K ) ( 0. `  K
)  /\  p ( le `  K ) x )  /\  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) )  ->  p  =/=  q )
2423reximi 2726 . . . . . 6  |-  ( E. q  e.  A  ( ( -.  p ( le `  K ) ( 0. `  K
)  /\  p ( le `  K ) x )  /\  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) )  ->  E. q  e.  A  p  =/=  q )
2524reximi 2726 . . . . 5  |-  ( E. p  e.  A  E. q  e.  A  (
( -.  p ( le `  K ) ( 0. `  K
)  /\  p ( le `  K ) x )  /\  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) )  ->  E. p  e.  A  E. q  e.  A  p  =/=  q )
2621, 25sylbir 204 . . . 4  |-  ( ( E. p  e.  A  ( -.  p ( le `  K ) ( 0. `  K )  /\  p ( le
`  K ) x )  /\  E. q  e.  A  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) )  ->  E. p  e.  A  E. q  e.  A  p  =/=  q )
2720, 26syl6 29 . . 3  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( ( ( 0.
`  K ) ( lt `  K ) x  /\  x ( lt `  K ) ( 1. `  K
) )  ->  E. p  e.  A  E. q  e.  A  p  =/=  q ) )
2827rexlimdva 2743 . 2  |-  ( K  e.  HL  ->  ( E. x  e.  ( Base `  K ) ( ( 0. `  K
) ( lt `  K ) x  /\  x ( lt `  K ) ( 1.
`  K ) )  ->  E. p  e.  A  E. q  e.  A  p  =/=  q ) )
295, 28mpd 14 1  |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  p  =/=  q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   class class class wbr 4102   ` cfv 5334   Basecbs 13239   lecple 13306   ltcplt 14168   0.cp0 14236   1.cp1 14237   OPcops 29414   Atomscatm 29505   HLchlt 29592
This theorem is referenced by:  atex  29647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593
  Copyright terms: Public domain W3C validator