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

Theorem cdlemn6 30543
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
StepHypRef Expression
1 simp1 960 . . 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 988 . . . 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 29359 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
81, 7syl 17 . . . . 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 986 . . . . 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 29919 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
131, 8, 9, 12syl3anc 1187 . . . 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 30107 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( s `  F )  e.  T
)
161, 2, 13, 15syl3anc 1187 . . 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 989 . . 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 30130 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
211, 20syl 17 . . 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 2256 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
24 cdlemn8.s . . . 4  |-  .+  =  ( +g  `  U )
25 eqid 2256 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
265, 10, 14, 22, 23, 24, 25dvhopvadd 30434 . . 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 1196 . 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 2256 . . . . . . 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 30425 . . . . . 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 17 . . . . 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 5795 . . . 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 30132 . . . . 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 645 . . . 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 2288 . . 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 3763 . 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 2288 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   <.cop 3603   class class class wbr 3983    e. cmpt 4037    _I cid 4262    |` cres 4649    o. ccom 4651   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   iota_crio 6249   Basecbs 13096   +g cplusg 13156  Scalarcsca 13159   lecple 13163   occoc 13164   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   TEndoctendo 30092   DVecHcdvh 30419
This theorem is referenced by:  cdlemn7  30544  dihordlem6  30554
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 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-n0 9919  df-z 9978  df-uz 10184  df-fz 10735  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-plusg 13169  df-mulr 13170  df-sca 13172  df-vsca 13173  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499  df-tendo 30095  df-edring 30097  df-dvech 30420
  Copyright terms: Public domain W3C validator