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Theorem hl2at 29594
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 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 eqid 2283 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2283 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
51, 2, 3, 4hlhgt2 29578 . 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 29552 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
87adantr 451 . . . . . . 7  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
91, 3op0cl 29374 . . . . . . 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 2283 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
13 hl2atom.a . . . . . . 7  |-  A  =  ( Atoms `  K )
141, 12, 2, 13hlrelat1 29589 . . . . . 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 29375 . . . . . . 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 29589 . . . . . 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 2707 . . . . 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 4041 . . . . . . . 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 2650 . . . . . 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 2650 . . . . 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 2667 . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   ltcplt 14075   0.cp0 14143   1.cp1 14144   OPcops 29362   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  atex  29595
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-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
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