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Theorem cdlemg18 36805
Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg18  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  W )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r
Allowed substitution hints:    R( r)    T( r)    H( r)    K( r)    ./\ ( r)

Proof of Theorem cdlemg18
StepHypRef Expression
1 simp11 1024 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp21r 1112 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  G  e.  T )
3 simp12 1025 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
4 cdlemg12.l . . 3  |-  .<_  =  ( le `  K )
5 cdlemg12.j . . 3  |-  .\/  =  ( join `  K )
6 cdlemg12.m . . 3  |-  ./\  =  ( meet `  K )
7 cdlemg12.a . . 3  |-  A  =  ( Atoms `  K )
8 cdlemg12.h . . 3  |-  H  =  ( LHyp `  K
)
9 cdlemg12.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
10 cdlemg12b.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
114, 5, 6, 7, 8, 9, 10cdlemg18d 36804 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  A )
12 simp23 1029 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( G `  P
)  =/=  P )
13 simp1 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
14 simp21l 1111 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  F  e.  T )
15 simp22 1028 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  P  =/=  Q )
16 simp31 1030 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( R `  G
)  .<_  ( P  .\/  Q ) )
17 simp33 1032 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
184, 5, 6, 7, 8, 9, 10cdlemg17 36800 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( ( G `  P )  =/=  P  /\  ( R `
 G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( G `  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) ) )  =  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) ) )
1913, 14, 2, 15, 12, 16, 17, 18syl133anc 1249 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( G `  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) ) )  =  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) ) )
204, 7, 8, 9ltrnatlw 36305 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  A )  /\  (
( G `  P
)  =/=  P  /\  ( G `  ( ( P  .\/  ( F `
 ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) ) )  =  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  .<_  W )
211, 2, 3, 11, 12, 19, 20syl132anc 1244 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   meetcmee 15773   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222   trLctrl 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281
This theorem is referenced by:  cdlemg19a  36806
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