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Theorem lneq2at 32776
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 1027 . . . 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 1028 . . . 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 530 . . 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 1029 . . 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 32774 . . . 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 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 1026 . . . . 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 1126 . . . . . . . 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 1129 . . . . . . . . 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 1130 . . . . . . . . 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 530 . . . . . . . 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 998 . . . . . . . 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 1177 . . . . . . 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 1131 . . . . . . 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 530 . . . . . 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 32362 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
231, 22syl 17 . . . . . . . . . 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 1030 . . . . . . . . . . . 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 32288 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  B )
2624, 25syl 17 . . . . . . . . . . 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 1031 . . . . . . . . . . . 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 32288 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  B )
2927, 28syl 17 . . . . . . . . . . 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 1177 . . . . . . . . . 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 530 . . . . . . . . 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 999 . . . . . . . . 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 15908 . . . . . . . . . 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 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 1018 . . . . . . 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 4418 . . . . . 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 1000 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  K  e.  HL )
40 simpl2l 1050 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  e.  A )
41 simpl2r 1051 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  Q  e.  A )
42 simpr 459 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  =/=  Q )
43 simpl3 1002 . . . . . . . 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 32475 . . . . . . . 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 1243 . . . . . . 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 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 2446 . . . 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 1196 . . 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 2901 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   Latclat 15891   Atomscatm 32262   HLchlt 32349   Linesclines 32492   pmapcpmap 32495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-lines 32499  df-pmap 32502
This theorem is referenced by:  lnjatN  32778  lncmp  32781  cdlema1N  32789
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