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Theorem ps-2c 30014
Description: Variation of projective geometry axiom ps-2 29964. (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 987 . 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 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 ) ) ) )  ->  P  e.  A )
3 simp21 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 ) ) ) )  ->  R  e.  A )
4 hllat 29850 . . . . 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 2408 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 29776 . . . . 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 989 . . . . 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 29776 . . . . 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 29776 . . . . 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 1080 . . . 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 14458 . . . 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 1197 . . 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 2408 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
2117, 7, 20llni2 29998 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .\/  R )  e.  (
LLines `  K ) )
221, 2, 3, 19, 21syl31anc 1187 . 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 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 ) ) ) )  ->  S  e.  A )
24 simp23 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 ) ) ) )  ->  T  e.  A )
25 simp31r 1081 . . 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 29998 . . 3  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
271, 23, 24, 25, 26syl31anc 1187 . 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 994 . 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 995 . . 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 2408 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
3216, 17, 30, 31, 7ps-2b 29968 . . 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 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 )  ./\  ( S  .\/  T ) )  =/=  ( 0. `  K
) )
3430, 31, 7, 202llnmat 30010 . 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 1192 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   0.cp0 14425   Latclat 14433   Atomscatm 29750   HLchlt 29837   LLinesclln 29977
This theorem is referenced by:  cdlemg18c  31166  dia2dimlem1  31551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984
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