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Theorem lneq2at 33728
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 1018 . . . 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 1019 . . . 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 532 . . 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 1020 . . 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 33726 . . . 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 207 . . 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 60 . 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 1017 . . . . 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 1117 . . . . . . . 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 1120 . . . . . . . . 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 1121 . . . . . . . . 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 532 . . . . . . . 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 989 . . . . . . . 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 1168 . . . . . . 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 1122 . . . . . . 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 532 . . . . . 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 33314 . . . . . . . . . . 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 1021 . . . . . . . . . . . 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 33240 . . . . . . . . . . . 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 1022 . . . . . . . . . . . 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 33240 . . . . . . . . . . . 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 1168 . . . . . . . . . 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 532 . . . . . . . . 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 990 . . . . . . . . 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 15334 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  <->  ( P  .\/  Q )  .<_  X ) )
3534biimpd 207 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  -> 
( P  .\/  Q
)  .<_  X ) )
3631, 32, 35sylc 60 . . . . . . . 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 1009 . . . . . . 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 4414 . . . . . 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 991 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  K  e.  HL )
40 simpl2l 1041 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  e.  A )
41 simpl2r 1042 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  Q  e.  A )
42 simpr 461 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  =/=  Q )
43 simpl3 993 . . . . . . . 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 33427 . . . . . . . 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 1232 . . . . . . 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 207 . . . . . 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 60 . . . . 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 2495 . . . 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 1187 . . 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 2943 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   Latclat 15317   Atomscatm 33214   HLchlt 33301   Linesclines 33444   pmapcpmap 33447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-clat 15380  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-lines 33451  df-pmap 33454
This theorem is referenced by:  lnjatN  33730  lncmp  33733  cdlema1N  33741
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