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Theorem lnatexN 29968
Description: There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnatex.b  |-  B  =  ( Base `  K
)
lnatex.l  |-  .<_  =  ( le `  K )
lnatex.a  |-  A  =  ( Atoms `  K )
lnatex.n  |-  N  =  ( Lines `  K )
lnatex.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
lnatexN  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
Distinct variable groups:    A, q    .<_ , q    P, q    X, q
Allowed substitution hints:    B( q)    K( q)    M( q)    N( q)

Proof of Theorem lnatexN
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnatex.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2283 . . . 4  |-  ( join `  K )  =  (
join `  K )
3 lnatex.a . . . 4  |-  A  =  ( Atoms `  K )
4 lnatex.n . . . 4  |-  N  =  ( Lines `  K )
5 lnatex.m . . . 4  |-  M  =  ( pmap `  K
)
61, 2, 3, 4, 5isline3 29965 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. r  e.  A  E. s  e.  A  (
r  =/=  s  /\  X  =  ( r
( join `  K )
s ) ) ) )
76biimp3a 1281 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )
8 simpl2r 1009 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  s  e.  A )
9 simpl3l 1010 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  r  =/=  s )
109necomd 2529 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  s  =/=  r )
11 simpr 447 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  r  =  P )
1210, 11neeqtrd 2468 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  s  =/=  P )
13 simpl11 1030 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  K  e.  HL )
14 simpl2l 1008 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  r  e.  A )
15 lnatex.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1615, 2, 3hlatlej2 29565 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  s  .<_  ( r
( join `  K )
s ) )
1713, 14, 8, 16syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  s  .<_  ( r ( join `  K ) s ) )
18 simpl3r 1011 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  X  =  ( r (
join `  K )
s ) )
1917, 18breqtrrd 4049 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  s  .<_  X )
20 neeq1 2454 . . . . . . . 8  |-  ( q  =  s  ->  (
q  =/=  P  <->  s  =/=  P ) )
21 breq1 4026 . . . . . . . 8  |-  ( q  =  s  ->  (
q  .<_  X  <->  s  .<_  X ) )
2220, 21anbi12d 691 . . . . . . 7  |-  ( q  =  s  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( s  =/=  P  /\  s  .<_  X ) ) )
2322rspcev 2884 . . . . . 6  |-  ( ( s  e.  A  /\  ( s  =/=  P  /\  s  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
248, 12, 19, 23syl12anc 1180 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =  P )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
25 simpl2l 1008 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  r  e.  A )
26 simpr 447 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  r  =/=  P )
27 simpl11 1030 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  K  e.  HL )
28 simpl2r 1009 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  s  e.  A )
2915, 2, 3hlatlej1 29564 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  r  .<_  ( r
( join `  K )
s ) )
3027, 25, 28, 29syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  r  .<_  ( r ( join `  K ) s ) )
31 simpl3r 1011 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  X  =  ( r (
join `  K )
s ) )
3230, 31breqtrrd 4049 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  r  .<_  X )
33 neeq1 2454 . . . . . . . 8  |-  ( q  =  r  ->  (
q  =/=  P  <->  r  =/=  P ) )
34 breq1 4026 . . . . . . . 8  |-  ( q  =  r  ->  (
q  .<_  X  <->  r  .<_  X ) )
3533, 34anbi12d 691 . . . . . . 7  |-  ( q  =  r  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( r  =/=  P  /\  r  .<_  X ) ) )
3635rspcev 2884 . . . . . 6  |-  ( ( r  e.  A  /\  ( r  =/=  P  /\  r  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
3725, 26, 32, 36syl12anc 1180 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  (
r  e.  A  /\  s  e.  A )  /\  ( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) ) )  /\  r  =/= 
P )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
3824, 37pm2.61dane 2524 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( r  e.  A  /\  s  e.  A
)  /\  ( r  =/=  s  /\  X  =  ( r ( join `  K ) s ) ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
39383exp 1150 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  -> 
( ( r  e.  A  /\  s  e.  A )  ->  (
( r  =/=  s  /\  X  =  (
r ( join `  K
) s ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) ) ) )
4039rexlimdvv 2673 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  -> 
( E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  X  =  ( r ( join `  K ) s ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) ) )
417, 40mpd 14 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   Linesclines 29683   pmapcpmap 29686
This theorem is referenced by:  lnjatN  29969
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  df-lines 29690  df-pmap 29693
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