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Theorem ps-2c 32545
Description: Variation of projective geometry axiom ps-2 32495. (Contributed by NM, 3-Jul-2012.)
Hypotheses
Ref Expression
2atm.l  |-  .<_  =  ( le `  K )
2atm.j  |-  .\/  =  ( join `  K )
2atm.m  |-  ./\  =  ( meet `  K )
2atm.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
ps-2c  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  A )

Proof of Theorem ps-2c
StepHypRef Expression
1 simp11 1027 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  K  e.  HL )
2 simp12 1028 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  P  e.  A )
3 simp21 1030 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  R  e.  A )
4 hllat 32381 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  K  e.  Lat )
6 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 32307 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
92, 8syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  P  e.  ( Base `  K ) )
10 simp13 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  Q  e.  A )
116, 7atbase 32307 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 11syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  Q  e.  ( Base `  K ) )
136, 7atbase 32307 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
143, 13syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  R  e.  ( Base `  K ) )
15 simp31l 1120 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
16 2atm.l . . . . 5  |-  .<_  =  ( le `  K )
17 2atm.j . . . . 5  |-  .\/  =  ( join `  K )
186, 16, 17latnlej1r 16024 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  /\  -.  P  .<_  ( Q  .\/  R ) )  ->  P  =/=  R )
195, 9, 12, 14, 15, 18syl131anc 1243 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  P  =/=  R )
20 eqid 2402 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
2117, 7, 20llni2 32529 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .\/  R )  e.  (
LLines `  K ) )
221, 2, 3, 19, 21syl31anc 1233 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( P  .\/  R
)  e.  ( LLines `  K ) )
23 simp22 1031 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  e.  A )
24 simp23 1032 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  T  e.  A )
25 simp31r 1121 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  =/=  T )
2617, 7, 20llni2 32529 . . 3  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
271, 23, 24, 25, 26syl31anc 1233 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  .\/  T
)  e.  ( LLines `  K ) )
28 simp32 1034 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( P  .\/  R
)  =/=  ( S 
.\/  T ) )
29 simp33 1035 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) )
30 2atm.m . . . 4  |-  ./\  =  ( meet `  K )
31 eqid 2402 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
3216, 17, 30, 31, 7ps-2b 32499 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/=  ( 0. `  K
) )
331, 2, 10, 3, 23, 24, 15, 25, 29, 32syl333anc 1262 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/=  ( 0. `  K
) )
3430, 31, 7, 202llnmat 32541 . 2  |-  ( ( ( K  e.  HL  /\  ( P  .\/  R
)  e.  ( LLines `  K )  /\  ( S  .\/  T )  e.  ( LLines `  K )
)  /\  ( ( P  .\/  R )  =/=  ( S  .\/  T
)  /\  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/=  ( 0.
`  K ) ) )  ->  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  A )
351, 22, 27, 28, 33, 34syl32anc 1238 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( ( -.  P  .<_  ( Q 
.\/  R )  /\  S  =/=  T )  /\  ( P  .\/  R )  =/=  ( S  .\/  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   0.cp0 15991   Latclat 15999   Atomscatm 32281   HLchlt 32368   LLinesclln 32508
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
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 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515
This theorem is referenced by:  cdlemg18c  33699  dia2dimlem1  34084
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