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Theorem cdlemg28a 33980
Description: Part of proof of Lemma G of [Crawley] p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-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
cdlemg28a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( z 
.\/  ( F `  ( G `  z ) ) )  ./\  W
) )

Proof of Theorem cdlemg28a
StepHypRef Expression
1 simp11 1035 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp12 1036 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
3 simp21 1038 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( z  e.  A  /\  -.  z  .<_  W ) )
4 simp22 1039 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  ->  F  e.  T )
5 simp23 1040 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  ->  G  e.  T )
6 simp1 1005 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) ) )
7 simp21l 1122 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
z  e.  A )
8 simp31l 1128 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
v  =/=  ( R `
 F ) )
9 simp32 1042 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
z  .<_  ( P  .\/  v ) )
10 simp33l 1132 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( F `  P
)  =/=  P )
11 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
12 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
13 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
14 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
15 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
16 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
17 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
1811, 12, 13, 14, 15, 16, 17cdlemg27a 33979 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  -.  ( R `  F ) 
.<_  ( P  .\/  z
) )
196, 7, 4, 8, 9, 10, 18syl123anc 1281 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  ->  -.  ( R `  F
)  .<_  ( P  .\/  z ) )
20 simp31r 1129 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
v  =/=  ( R `
 G ) )
21 simp33r 1133 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( G `  P
)  =/=  P )
2211, 12, 13, 14, 15, 16, 17cdlemg27a 33979 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  G  e.  T
)  /\  ( v  =/=  ( R `  G
)  /\  z  .<_  ( P  .\/  v )  /\  ( G `  P )  =/=  P
) )  ->  -.  ( R `  G ) 
.<_  ( P  .\/  z
) )
236, 7, 5, 20, 9, 21, 22syl123anc 1281 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  ->  -.  ( R `  G
)  .<_  ( P  .\/  z ) )
2411, 12, 13, 14, 15, 16, 17cdlemg25zz 33977 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  ( R `  F )  .<_  ( P  .\/  z
)  /\  -.  ( R `  G )  .<_  ( P  .\/  z
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( z 
.\/  ( F `  ( G `  z ) ) )  ./\  W
) )
251, 2, 3, 4, 5, 19, 23, 24syl133anc 1287 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( z 
.\/  ( F `  ( G `  z ) ) )  ./\  W
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15160   joincjn 16144   meetcmee 16145   Atomscatm 32549   HLchlt 32636   LHypclh 33269   LTrncltrn 33386   trLctrl 33444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-riotaBAD 32245
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7028  df-map 7482  df-preset 16128  df-poset 16146  df-plt 16159  df-lub 16175  df-glb 16176  df-join 16177  df-meet 16178  df-p0 16240  df-p1 16241  df-lat 16247  df-clat 16309  df-oposet 32462  df-ol 32464  df-oml 32465  df-covers 32552  df-ats 32553  df-atl 32584  df-cvlat 32608  df-hlat 32637  df-llines 32783  df-lplanes 32784  df-lvols 32785  df-lines 32786  df-psubsp 32788  df-pmap 32789  df-padd 33081  df-lhyp 33273  df-laut 33274  df-ldil 33389  df-ltrn 33390  df-trl 33445
This theorem is referenced by:  cdlemg28  33991
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