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Theorem ps-2c 29786
Description: Variation of projective geometry axiom ps-2 29736. (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 985 . 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 986 . . 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 988 . . 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 29622 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 15 . . . 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 2358 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 29548 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
92, 8syl 15 . . . 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 987 . . . . 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 29548 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 11syl 15 . . . 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 29548 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
143, 13syl 15 . . . 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 1078 . . . 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 14275 . . . 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 1195 . . 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 2358 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
2117, 7, 20llni2 29770 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .\/  R )  e.  (
LLines `  K ) )
221, 2, 3, 19, 21syl31anc 1185 . 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 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 ) ) ) )  ->  S  e.  A )
24 simp23 990 . . 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 1079 . . 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 29770 . . 3  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
271, 23, 24, 25, 26syl31anc 1185 . 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 992 . 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 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 ) ) ) )  -> 
( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) )
30 2atm.m . . . 4  |-  ./\  =  ( meet `  K )
31 eqid 2358 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
3216, 17, 30, 31, 7ps-2b 29740 . . 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 1214 . 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 29782 . 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 1190 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   lecple 13312   joincjn 14177   meetcmee 14178   0.cp0 14242   Latclat 14250   Atomscatm 29522   HLchlt 29609   LLinesclln 29749
This theorem is referenced by:  cdlemg18c  30938  dia2dimlem1  31323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-lat 14251  df-clat 14313  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-llines 29756
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