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Theorem lnatexN 30514
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 2436 . . . 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 30511 . . 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 2682 . . . . . . 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 2621 . . . . . 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 30111 . . . . . . . 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 4231 . . . . . 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 2607 . . . . . . . 8  |-  ( q  =  s  ->  (
q  =/=  P  <->  s  =/=  P ) )
21 breq1 4208 . . . . . . . 8  |-  ( q  =  s  ->  (
q  .<_  X  <->  s  .<_  X ) )
2220, 21anbi12d 692 . . . . . . 7  |-  ( q  =  s  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( s  =/=  P  /\  s  .<_  X ) ) )
2322rspcev 3045 . . . . . 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 30110 . . . . . . . 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 4231 . . . . . 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 2607 . . . . . . . 8  |-  ( q  =  r  ->  (
q  =/=  P  <->  r  =/=  P ) )
34 breq1 4208 . . . . . . . 8  |-  ( q  =  r  ->  (
q  .<_  X  <->  r  .<_  X ) )
3533, 34anbi12d 692 . . . . . . 7  |-  ( q  =  r  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( r  =/=  P  /\  r  .<_  X ) ) )
3635rspcev 3045 . . . . . 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 2677 . . . 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 2829 . 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 1652    e. wcel 1725    =/= wne 2599   E.wrex 2699   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   Basecbs 13462   lecple 13529   joincjn 14394   Atomscatm 29999   HLchlt 30086   Linesclines 30229   pmapcpmap 30232
This theorem is referenced by:  lnjatN  30515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-lines 30236  df-pmap 30239
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