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Theorem hl2at 30276
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 2438 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2438 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 eqid 2438 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2438 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
51, 2, 3, 4hlhgt2 30260 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) ( ( 0.
`  K ) ( lt `  K ) x  /\  x ( lt `  K ) ( 1. `  K
) ) )
6 simpl 445 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  HL )
7 hlop 30234 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
87adantr 453 . . . . . . 7  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
91, 3op0cl 30056 . . . . . . 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 449 . . . . . 6  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  x  e.  ( Base `  K ) )
12 eqid 2438 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
13 hl2atom.a . . . . . . 7  |-  A  =  ( Atoms `  K )
141, 12, 2, 13hlrelat1 30271 . . . . . 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 1185 . . . . 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 30057 . . . . . . 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 30271 . . . . . 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 1281 . . . . 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 548 . . . 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 2877 . . . . 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 4233 . . . . . . . 8  |-  ( ( p ( le `  K ) x  /\  -.  q ( le `  K ) x )  ->  p  =/=  q
)
2322ad2ant2lr 730 . . . . . . 7  |-  ( ( ( -.  p ( le `  K ) ( 0. `  K
)  /\  p ( le `  K ) x )  /\  ( -.  q ( le `  K ) x  /\  q ( le `  K ) ( 1.
`  K ) ) )  ->  p  =/=  q )
2423reximi 2815 . . . . . 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 2815 . . . . 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 206 . . . 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 32 . . 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 2832 . 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 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   ltcplt 14403   0.cp0 14471   1.cp1 14472   OPcops 30044   Atomscatm 30135   HLchlt 30222
This theorem is referenced by:  atex  30277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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