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Theorem ps-2c 30399
Description: Variation of projective geometry axiom ps-2 30349. (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 988 . 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 989 . . 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 991 . . 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 30235 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 16 . . . 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 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 30161 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
92, 8syl 16 . . . 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 990 . . . . 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 30161 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 11syl 16 . . . 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 30161 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
143, 13syl 16 . . . 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 1081 . . . 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 14504 . . . 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 1198 . . 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 2438 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
2117, 7, 20llni2 30383 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .\/  R )  e.  (
LLines `  K ) )
221, 2, 3, 19, 21syl31anc 1188 . 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 992 . . 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 993 . . 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 1082 . . 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 30383 . . 3  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
271, 23, 24, 25, 26syl31anc 1188 . 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 995 . 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 996 . . 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 2438 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
3216, 17, 30, 31, 7ps-2b 30353 . . 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 1217 . 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 30395 . 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 1193 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   0.cp0 14471   Latclat 14479   Atomscatm 30135   HLchlt 30222   LLinesclln 30362
This theorem is referenced by:  cdlemg18c  31551  dia2dimlem1  31936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369
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