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Theorem cdlemg28a 30049
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 30048 . . 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 30048 . . 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 30046 . 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 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   lecple 13177   joincjn 14040   meetcmee 14041   Atomscatm 28620   HLchlt 28707   LHypclh 29340   LTrncltrn 29457   trLctrl 29514
This theorem is referenced by:  cdlemg28  30060
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515
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