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Theorem lnatexN 29945
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 2381 . . . 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 29942 . . 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 1283 . 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 1011 . . . . . 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 1012 . . . . . . . 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 2627 . . . . . . 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 448 . . . . . . 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 2566 . . . . . 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 1032 . . . . . . . 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 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  e.  A )
15 lnatex.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1615, 2, 3hlatlej2 29542 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  s  .<_  ( r
( join `  K )
s ) )
1713, 14, 8, 16syl3anc 1184 . . . . . . 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 1013 . . . . . . 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 4173 . . . . . 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 2552 . . . . . . . 8  |-  ( q  =  s  ->  (
q  =/=  P  <->  s  =/=  P ) )
21 breq1 4150 . . . . . . . 8  |-  ( q  =  s  ->  (
q  .<_  X  <->  s  .<_  X ) )
2220, 21anbi12d 692 . . . . . . 7  |-  ( q  =  s  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( s  =/=  P  /\  s  .<_  X ) ) )
2322rspcev 2989 . . . . . 6  |-  ( ( s  e.  A  /\  ( s  =/=  P  /\  s  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
248, 12, 19, 23syl12anc 1182 . . . . 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 1010 . . . . . 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 448 . . . . . 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 1032 . . . . . . . 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 1011 . . . . . . . 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 29541 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  r  .<_  ( r
( join `  K )
s ) )
3027, 25, 28, 29syl3anc 1184 . . . . . . 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 1013 . . . . . . 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 4173 . . . . . 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 2552 . . . . . . . 8  |-  ( q  =  r  ->  (
q  =/=  P  <->  r  =/=  P ) )
34 breq1 4150 . . . . . . . 8  |-  ( q  =  r  ->  (
q  .<_  X  <->  r  .<_  X ) )
3533, 34anbi12d 692 . . . . . . 7  |-  ( q  =  r  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( r  =/=  P  /\  r  .<_  X ) ) )
3635rspcev 2989 . . . . . 6  |-  ( ( r  e.  A  /\  ( r  =/=  P  /\  r  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
3725, 26, 32, 36syl12anc 1182 . . . . 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 2622 . . . 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 1152 . . 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 2773 . 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 15 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2544   E.wrex 2644   class class class wbr 4147   ` cfv 5388  (class class class)co 6014   Basecbs 13390   lecple 13457   joincjn 14322   Atomscatm 29430   HLchlt 29517   Linesclines 29660   pmapcpmap 29663
This theorem is referenced by:  lnjatN  29946
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-undef 6473  df-riota 6479  df-poset 14324  df-plt 14336  df-lub 14352  df-glb 14353  df-join 14354  df-meet 14355  df-p0 14389  df-lat 14396  df-clat 14458  df-oposet 29343  df-ol 29345  df-oml 29346  df-covers 29433  df-ats 29434  df-atl 29465  df-cvlat 29489  df-hlat 29518  df-lines 29667  df-pmap 29670
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