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Theorem hl2at 29887
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 2404 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2404 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 eqid 2404 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2404 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
51, 2, 3, 4hlhgt2 29871 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) ( ( 0.
`  K ) ( lt `  K ) x  /\  x ( lt `  K ) ( 1. `  K
) ) )
6 simpl 444 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  HL )
7 hlop 29845 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
87adantr 452 . . . . . . 7  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
91, 3op0cl 29667 . . . . . . 7  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
108, 9syl 16 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( 0. `  K
)  e.  ( Base `  K ) )
11 simpr 448 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  x  e.  ( Base `  K ) )
12 eqid 2404 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
13 hl2atom.a . . . . . . 7  |-  A  =  ( Atoms `  K )
141, 12, 2, 13hlrelat1 29882 . . . . . 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 1184 . . . . 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 29668 . . . . . . 7  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
178, 16syl 16 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( 1. `  K
)  e.  ( Base `  K ) )
181, 12, 2, 13hlrelat1 29882 . . . . . 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 1280 . . . . 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 547 . . . 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 2835 . . . . 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 4190 . . . . . . . 8  |-  ( ( p ( le `  K ) x  /\  -.  q ( le `  K ) x )  ->  p  =/=  q
)
2322ad2ant2lr 729 . . . . . . 7  |-  ( ( ( -.  p ( le `  K ) ( 0. `  K
)  /\  p ( le `  K ) x )  /\  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) )  ->  p  =/=  q )
2423reximi 2773 . . . . . 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 2773 . . . . 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 205 . . . 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 31 . . 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 2790 . 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 15 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 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   ltcplt 14353   0.cp0 14421   1.cp1 14422   OPcops 29655   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  atex  29888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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