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Theorem lnatexN 33053
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 2429 . . . 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 33050 . . 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 1364 . 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 1059 . . . . . 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 1060 . . . . . . . 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 2702 . . . . . . 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 462 . . . . . . 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 2726 . . . . . 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 1080 . . . . . . . 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 1058 . . . . . . . 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 32650 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  s  .<_  ( r
( join `  K )
s ) )
1713, 14, 8, 16syl3anc 1264 . . . . . . 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 1061 . . . . . . 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 4452 . . . . . 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 2712 . . . . . . . 8  |-  ( q  =  s  ->  (
q  =/=  P  <->  s  =/=  P ) )
21 breq1 4429 . . . . . . . 8  |-  ( q  =  s  ->  (
q  .<_  X  <->  s  .<_  X ) )
2220, 21anbi12d 715 . . . . . . 7  |-  ( q  =  s  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( s  =/=  P  /\  s  .<_  X ) ) )
2322rspcev 3188 . . . . . 6  |-  ( ( s  e.  A  /\  ( s  =/=  P  /\  s  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
248, 12, 19, 23syl12anc 1262 . . . . 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 1058 . . . . . 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 462 . . . . . 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 1080 . . . . . . . 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 1059 . . . . . . . 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 32649 . . . . . . . 8  |-  ( ( K  e.  HL  /\  r  e.  A  /\  s  e.  A )  ->  r  .<_  ( r
( join `  K )
s ) )
3027, 25, 28, 29syl3anc 1264 . . . . . . 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 1061 . . . . . . 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 4452 . . . . . 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 2712 . . . . . . . 8  |-  ( q  =  r  ->  (
q  =/=  P  <->  r  =/=  P ) )
34 breq1 4429 . . . . . . . 8  |-  ( q  =  r  ->  (
q  .<_  X  <->  r  .<_  X ) )
3533, 34anbi12d 715 . . . . . . 7  |-  ( q  =  r  ->  (
( q  =/=  P  /\  q  .<_  X )  <-> 
( r  =/=  P  /\  r  .<_  X ) ) )
3635rspcev 3188 . . . . . 6  |-  ( ( r  e.  A  /\  ( r  =/=  P  /\  r  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
3725, 26, 32, 36syl12anc 1262 . . . . 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 2749 . . . 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 1204 . . 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 2930 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Atomscatm 32538   HLchlt 32625   Linesclines 32768   pmapcpmap 32771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-lines 32775  df-pmap 32778
This theorem is referenced by:  lnjatN  33054
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