Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemn6 Structured version   Unicode version

Theorem cdlemn6 31901
Description: Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)
Hypotheses
Ref Expression
cdlemn8.b  |-  B  =  ( Base `  K
)
cdlemn8.l  |-  .<_  =  ( le `  K )
cdlemn8.a  |-  A  =  ( Atoms `  K )
cdlemn8.h  |-  H  =  ( LHyp `  K
)
cdlemn8.p  |-  P  =  ( ( oc `  K ) `  W
)
cdlemn8.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
cdlemn8.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn8.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdlemn8.u  |-  U  =  ( ( DVecH `  K
) `  W )
cdlemn8.s  |-  .+  =  ( +g  `  U )
cdlemn8.f  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
Assertion
Ref Expression
cdlemn6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( <. ( s `  F ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  F
)  o.  g ) ,  s >. )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    T, h    P, h    Q, h    h, W
Allowed substitution hints:    A( g, s)    B( g, s)    P( g, s)    .+ ( g, h, s)    Q( g, s)    R( g, h, s)    T( g, s)    U( g, h, s)    E( g, h, s)    F( g, h, s)    H( g, s)    K( g, s)    .<_ ( g, s)    O( g, h, s)    W( g, s)

Proof of Theorem cdlemn6
Dummy variables  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp3l 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
s  e.  E )
3 cdlemn8.l . . . . . . 7  |-  .<_  =  ( le `  K )
4 cdlemn8.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 cdlemn8.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
6 cdlemn8.p . . . . . . 7  |-  P  =  ( ( oc `  K ) `  W
)
73, 4, 5, 6lhpocnel2 30717 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
81, 7syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
9 simp2l 983 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
10 cdlemn8.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
11 cdlemn8.f . . . . . 6  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
123, 4, 5, 10, 11ltrniotacl 31277 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
131, 8, 9, 12syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  ->  F  e.  T )
14 cdlemn8.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
155, 10, 14tendocl 31465 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( s `  F )  e.  T
)
161, 2, 13, 15syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s `  F
)  e.  T )
17 simp3r 986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
g  e.  T )
18 cdlemn8.b . . . . 5  |-  B  =  ( Base `  K
)
19 cdlemn8.o . . . . 5  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
2018, 5, 10, 14, 19tendo0cl 31488 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
211, 20syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  ->  O  e.  E )
22 cdlemn8.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
23 eqid 2435 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
24 cdlemn8.s . . . 4  |-  .+  =  ( +g  `  U )
25 eqid 2435 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
265, 10, 14, 22, 23, 24, 25dvhopvadd 31792 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s `
 F )  e.  T  /\  s  e.  E )  /\  (
g  e.  T  /\  O  e.  E )
)  ->  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )  =  <. ( ( s `
 F )  o.  g ) ,  ( s ( +g  `  (Scalar `  U ) ) O ) >. )
271, 16, 2, 17, 21, 26syl122anc 1193 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( <. ( s `  F ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  F
)  o.  g ) ,  ( s ( +g  `  (Scalar `  U ) ) O ) >. )
28 eqid 2435 . . . . . . 7  |-  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) )  =  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) )
295, 10, 14, 22, 23, 28, 25dvhfplusr 31783 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  (Scalar `  U ) )  =  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  (
u `  h )
) ) ) )
301, 29syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( +g  `  (Scalar `  U ) )  =  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  (
u `  h )
) ) ) )
3130oveqd 6090 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s ( +g  `  (Scalar `  U )
) O )  =  ( s ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) ) O ) )
3218, 5, 10, 14, 19, 28tendo0plr 31490 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E
)  ->  ( s
( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  (
u `  h )
) ) ) O )  =  s )
331, 2, 32syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) ) O )  =  s )
3431, 33eqtrd 2467 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s ( +g  `  (Scalar `  U )
) O )  =  s )
3534opeq2d 3983 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  ->  <. ( ( s `  F )  o.  g
) ,  ( s ( +g  `  (Scalar `  U ) ) O ) >.  =  <. ( ( s `  F
)  o.  g ) ,  s >. )
3627, 35eqtrd 2467 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( <. ( s `  F ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  F
)  o.  g ) ,  s >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    e. cmpt 4258    _I cid 4485    |` cres 4872    o. ccom 4874   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   iota_crio 6534   Basecbs 13459   +g cplusg 13519  Scalarcsca 13522   lecple 13526   occoc 13527   Atomscatm 29962   HLchlt 30049   LHypclh 30682   LTrncltrn 30799   TEndoctendo 31450   DVecHcdvh 31777
This theorem is referenced by:  cdlemn7  31902  dihordlem6  31912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197  df-lvols 30198  df-lines 30199  df-psubsp 30201  df-pmap 30202  df-padd 30494  df-lhyp 30686  df-laut 30687  df-ldil 30802  df-ltrn 30803  df-trl 30857  df-tendo 31453  df-edring 31455  df-dvech 31778
  Copyright terms: Public domain W3C validator