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Theorem lneq2at 29967
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 985 . . . 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 986 . . . 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 518 . . 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 987 . . 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 29965 . . . 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 198 . . 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 56 . 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 984 . . . . 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 1084 . . . . . . . 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 1087 . . . . . . . . 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 1088 . . . . . . . . 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 518 . . . . . . . 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 956 . . . . . . . 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 1132 . . . . . . 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 1089 . . . . . . 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 518 . . . . . 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 29553 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
231, 22syl 15 . . . . . . . . . 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 988 . . . . . . . . . . . 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 29479 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  B )
2624, 25syl 15 . . . . . . . . . . 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 989 . . . . . . . . . . . 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 29479 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  B )
2927, 28syl 15 . . . . . . . . . . 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 1132 . . . . . . . . . 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 518 . . . . . . . . 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 957 . . . . . . . . 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 14168 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  <->  ( P  .\/  Q )  .<_  X ) )
3534biimpd 198 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  -> 
( P  .\/  Q
)  .<_  X ) )
3631, 32, 35sylc 56 . . . . . . . 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 976 . . . . . . 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 4047 . . . . . 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 958 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  K  e.  HL )
40 simpl2l 1008 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  e.  A )
41 simpl2r 1009 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  Q  e.  A )
42 simpr 447 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( r  e.  A  /\  s  e.  A ) )  /\  P  =/=  Q )  ->  P  =/=  Q )
43 simpl3 960 . . . . . . . 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 29666 . . . . . . . 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 1195 . . . . . . 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 198 . . . . . 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 56 . . . . 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 2318 . . . 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 1150 . . 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 2673 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   Linesclines 29683   pmapcpmap 29686
This theorem is referenced by:  lnjatN  29969  lncmp  29972  cdlema1N  29980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lines 29690  df-pmap 29693
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