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

Theorem cdlemn6 35155
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 988 . . 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 1016 . . . 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 33971 . . . . . 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 1014 . . . . 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 34531 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
131, 8, 9, 12syl3anc 1219 . . . 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 34719 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( s `  F )  e.  T
)
161, 2, 13, 15syl3anc 1219 . . 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 1017 . . 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 34742 . . . 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 2451 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
24 cdlemn8.s . . . 4  |-  .+  =  ( +g  `  U )
25 eqid 2451 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
265, 10, 14, 22, 23, 24, 25dvhopvadd 35046 . . 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 1228 . 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 2451 . . . . . . 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 35037 . . . . . 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 6209 . . . 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 34744 . . . . 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 661 . . . 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 2492 . . 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 4166 . 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 2492 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   <.cop 3983   class class class wbr 4392    |-> cmpt 4450    _I cid 4731    |` cres 4942    o. ccom 4944   ` cfv 5518   iota_crio 6152  (class class class)co 6192    |-> cmpt2 6194   Basecbs 14278   +g cplusg 14342  Scalarcsca 14345   lecple 14349   occoc 14350   Atomscatm 33216   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   TEndoctendo 34704   DVecHcdvh 35031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-undef 6894  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111  df-tendo 34707  df-edring 34709  df-dvech 35032
This theorem is referenced by:  cdlemn7  35156  dihordlem6  35166
  Copyright terms: Public domain W3C validator