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Theorem cdlemg28a 29786
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 990 . 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 991 . 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 993 . 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 994 . 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 995 . 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 960 . . 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 1077 . . 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 1083 . . 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 997 . . 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 1087 . . 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 29785 . . 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 1204 . 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 1084 . . 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 1088 . . 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 29785 . . 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 1204 . 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 29783 . 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 1210 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 set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   HLchlt 28444   LHypclh 29077   LTrncltrn 29194   trLctrl 29251
This theorem is referenced by:  cdlemg28  29797
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593  df-lines 28594  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198  df-trl 29252
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