Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ps-2c Structured version   Unicode version

Theorem ps-2c 32894
Description: Variation of projective geometry axiom ps-2 32844. (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 1013 . 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 1014 . . 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 1016 . . 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 32730 . . . . 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 2441 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 32656 . . . . 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 1015 . . . . 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 32656 . . . . 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 32656 . . . . 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 1106 . . . 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 15236 . . . 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 1226 . . 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 2441 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
2117, 7, 20llni2 32878 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .\/  R )  e.  (
LLines `  K ) )
221, 2, 3, 19, 21syl31anc 1216 . 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 1017 . . 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 1018 . . 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 1107 . . 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 32878 . . 3  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
271, 23, 24, 25, 26syl31anc 1216 . 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 1020 . 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 1021 . . 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 2441 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
3216, 17, 30, 31, 7ps-2b 32848 . . 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 1245 . 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 32890 . 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 1221 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 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   meetcmee 15111   0.cp0 15203   Latclat 15211   Atomscatm 32630   HLchlt 32717   LLinesclln 32857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864
This theorem is referenced by:  cdlemg18c  34046  dia2dimlem1  34431
  Copyright terms: Public domain W3C validator