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Theorem lnatexN 33263
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 2438 . . . 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 33260 . . 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 1318 . 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 1042 . . . . . 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 1043 . . . . . . . 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 2690 . . . . . . 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 461 . . . . . . 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 2625 . . . . . 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 1063 . . . . . . . 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 1041 . . . . . . . 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 32860 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  s  .<_  ( r
( join `  K )
s ) )
1713, 14, 8, 16syl3anc 1218 . . . . . . 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 1044 . . . . . . 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 4313 . . . . . 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 2611 . . . . . . . 8  |-  ( q  =  s  ->  (
q  =/=  P  <->  s  =/=  P ) )
21 breq1 4290 . . . . . . . 8  |-  ( q  =  s  ->  (
q  .<_  X  <->  s  .<_  X ) )
2220, 21anbi12d 710 . . . . . . 7  |-  ( q  =  s  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( s  =/=  P  /\  s  .<_  X ) ) )
2322rspcev 3068 . . . . . 6  |-  ( ( s  e.  A  /\  ( s  =/=  P  /\  s  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
248, 12, 19, 23syl12anc 1216 . . . . 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 1041 . . . . . 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 461 . . . . . 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 1063 . . . . . . . 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 1042 . . . . . . . 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 32859 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  r  .<_  ( r
( join `  K )
s ) )
3027, 25, 28, 29syl3anc 1218 . . . . . . 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 1044 . . . . . . 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 4313 . . . . . 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 2611 . . . . . . . 8  |-  ( q  =  r  ->  (
q  =/=  P  <->  r  =/=  P ) )
34 breq1 4290 . . . . . . . 8  |-  ( q  =  r  ->  (
q  .<_  X  <->  r  .<_  X ) )
3533, 34anbi12d 710 . . . . . . 7  |-  ( q  =  r  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( r  =/=  P  /\  r  .<_  X ) ) )
3635rspcev 3068 . . . . . 6  |-  ( ( r  e.  A  /\  ( r  =/=  P  /\  r  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
3725, 26, 32, 36syl12anc 1216 . . . . 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 2684 . . . 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 1186 . . 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 2842 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   Atomscatm 32748   HLchlt 32835   Linesclines 32978   pmapcpmap 32981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lines 32985  df-pmap 32988
This theorem is referenced by:  lnjatN  33264
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