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Theorem lneq2at 30576
Description: A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
Hypotheses
Ref Expression
lneq2at.b  |-  B  =  ( Base `  K
)
lneq2at.l  |-  .<_  =  ( le `  K )
lneq2at.j  |-  .\/  =  ( join `  K )
lneq2at.a  |-  A  =  ( Atoms `  K )
lneq2at.n  |-  N  =  ( Lines `  K )
lneq2at.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
lneq2at  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  X  =  ( P  .\/  Q ) )

Proof of Theorem lneq2at
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 988 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  K  e.  HL )
2 simp12 989 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  X  e.  B )
31, 2jca 520 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  ( K  e.  HL  /\  X  e.  B ) )
4 simp13 990 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  ( M `  X )  e.  N )
5 lneq2at.b . . . . 5  |-  B  =  ( Base `  K
)
6 lneq2at.j . . . . 5  |-  .\/  =  ( join `  K )
7 lneq2at.a . . . . 5  |-  A  =  ( Atoms `  K )
8 lneq2at.n . . . . 5  |-  N  =  ( Lines `  K )
9 lneq2at.m . . . . 5  |-  M  =  ( pmap `  K
)
105, 6, 7, 8, 9isline3 30574 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. r  e.  A  E. s  e.  A  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) ) )
1110biimpd 200 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  ->  E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  X  =  (
r  .\/  s )
) ) )
123, 4, 11sylc 59 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  X  =  ( r  .\/  s
) ) )
13 simp3r 987 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  X  =  ( r  .\/  s ) )
14 simp111 1087 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  K  e.  HL )
15 simp121 1090 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  P  e.  A
)
16 simp122 1091 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  Q  e.  A
)
1715, 16jca 520 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  ( P  e.  A  /\  Q  e.  A ) )
18 simp2 959 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  ( r  e.  A  /\  s  e.  A ) )
1914, 17, 183jca 1135 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
r  e.  A  /\  s  e.  A )
) )
20 simp123 1092 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  P  =/=  Q
)
2119, 20jca 520 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( r  e.  A  /\  s  e.  A
) )  /\  P  =/=  Q ) )
22 hllat 30162 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
231, 22syl 16 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  K  e.  Lat )
24 simp21 991 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  P  e.  A )
255, 7atbase 30088 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  B )
2624, 25syl 16 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  P  e.  B )
27 simp22 992 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  Q  e.  A )
285, 7atbase 30088 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  B )
2927, 28syl 16 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  Q  e.  B )
3026, 29, 23jca 1135 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B )
)
3123, 30jca 520 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B )
) )
32 simp3 960 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  ( P  .<_  X  /\  Q  .<_  X ) )
33 lneq2at.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
345, 33, 6latjle12 14492 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  <->  ( P  .\/  Q )  .<_  X ) )
3534biimpd 200 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  -> 
( P  .\/  Q
)  .<_  X ) )
3631, 32, 35sylc 59 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  ( P  .\/  Q )  .<_  X )
37363ad2ant1 979 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  ( P  .\/  Q )  .<_  X )
3837, 13breqtrd 4237 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  ( P  .\/  Q )  .<_  ( r  .\/  s ) )
39 simpl1 961 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  K  e.  HL )
40 simpl2l 1011 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  e.  A )
41 simpl2r 1012 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  Q  e.  A )
42 simpr 449 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  =/=  Q )
43 simpl3 963 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  -> 
( r  e.  A  /\  s  e.  A
) )
4433, 6, 7ps-1 30275 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( r  e.  A  /\  s  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( r  .\/  s )  <->  ( P  .\/  Q )  =  ( r  .\/  s ) ) )
4539, 40, 41, 42, 43, 44syl131anc 1198 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  -> 
( ( P  .\/  Q )  .<_  ( r  .\/  s )  <->  ( P  .\/  Q )  =  ( r  .\/  s ) ) )
4645biimpd 200 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  -> 
( ( P  .\/  Q )  .<_  ( r  .\/  s )  ->  ( P  .\/  Q )  =  ( r  .\/  s
) ) )
4721, 38, 46sylc 59 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  ( P  .\/  Q )  =  ( r 
.\/  s ) )
4813, 47eqtr4d 2472 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `
 X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) )  /\  ( r  e.  A  /\  s  e.  A )  /\  (
r  =/=  s  /\  X  =  ( r  .\/  s ) ) )  ->  X  =  ( P  .\/  Q ) )
49483exp 1153 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  (
( r  e.  A  /\  s  e.  A
)  ->  ( (
r  =/=  s  /\  X  =  ( r  .\/  s ) )  ->  X  =  ( P  .\/  Q ) ) ) )
5049rexlimdvv 2837 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  ( E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  X  =  (
r  .\/  s )
)  ->  X  =  ( P  .\/  Q ) ) )
5112, 50mpd 15 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  .<_  X  /\  Q  .<_  X ) )  ->  X  =  ( P  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   Latclat 14475   Atomscatm 30062   HLchlt 30149   Linesclines 30292   pmapcpmap 30295
This theorem is referenced by:  lnjatN  30578  lncmp  30581  cdlema1N  30589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-lines 30299  df-pmap 30302
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