Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemn11a Structured version   Unicode version

Theorem cdlemn11a 34574
Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn11a.b  |-  B  =  ( Base `  K
)
cdlemn11a.l  |-  .<_  =  ( le `  K )
cdlemn11a.j  |-  .\/  =  ( join `  K )
cdlemn11a.a  |-  A  =  ( Atoms `  K )
cdlemn11a.h  |-  H  =  ( LHyp `  K
)
cdlemn11a.p  |-  P  =  ( ( oc `  K ) `  W
)
cdlemn11a.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
cdlemn11a.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn11a.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemn11a.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdlemn11a.i  |-  I  =  ( ( DIsoB `  K
) `  W )
cdlemn11a.J  |-  J  =  ( ( DIsoC `  K
) `  W )
cdlemn11a.u  |-  U  =  ( ( DVecH `  K
) `  W )
cdlemn11a.d  |-  .+  =  ( +g  `  U )
cdlemn11a.s  |-  .(+)  =  (
LSSum `  U )
cdlemn11a.f  |-  F  =  ( iota_ h  e.  T  ( h `  P
)  =  Q )
cdlemn11a.g  |-  G  =  ( iota_ h  e.  T  ( h `  P
)  =  N )
Assertion
Ref Expression
cdlemn11a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  <. G , 
(  _I  |`  T )
>.  e.  ( J `  N ) )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    h, N    P, h    Q, h    T, h   
h, W
Allowed substitution hints:    .+ ( h)    .(+) ( h)    R( h)    U( h)    E( h)    F( h)    G( h)    I( h)    J( h)    .\/ ( h)    O( h)    X( h)

Proof of Theorem cdlemn11a
StepHypRef Expression
1 simp1 983 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 cdlemn11a.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 cdlemn11a.a . . . . . . 7  |-  A  =  ( Atoms `  K )
4 cdlemn11a.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
5 cdlemn11a.p . . . . . . 7  |-  P  =  ( ( oc `  K ) `  W
)
62, 3, 4, 5lhpocnel2 33385 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
763ad2ant1 1004 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp22 1017 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( N  e.  A  /\  -.  N  .<_  W ) )
9 cdlemn11a.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
10 cdlemn11a.g . . . . . 6  |-  G  =  ( iota_ h  e.  T  ( h `  P
)  =  N )
112, 3, 4, 9, 10ltrniotacl 33945 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  ->  G  e.  T )
121, 7, 8, 11syl3anc 1213 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  G  e.  T )
13 tendospid 34384 . . . 4  |-  ( G  e.  T  ->  (
(  _I  |`  T ) `
 G )  =  G )
1412, 13syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( (  _I  |`  T ) `  G )  =  G )
1514eqcomd 2446 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  G  =  ( (  _I  |`  T ) `
 G ) )
16 cdlemn11a.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
174, 9, 16tendoidcl 34135 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
18173ad2ant1 1004 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  (  _I  |`  T )  e.  E
)
19 cdlemn11a.J . . . 4  |-  J  =  ( ( DIsoC `  K
) `  W )
20 riotaex 6053 . . . . 5  |-  ( iota_ h  e.  T  ( h `
 P )  =  N )  e.  _V
2110, 20eqeltri 2511 . . . 4  |-  G  e. 
_V
22 fvex 5698 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
239, 22eqeltri 2511 . . . . 5  |-  T  e. 
_V
24 resiexg 6513 . . . . 5  |-  ( T  e.  _V  ->  (  _I  |`  T )  e. 
_V )
2523, 24ax-mp 5 . . . 4  |-  (  _I  |`  T )  e.  _V
262, 3, 4, 5, 9, 16, 19, 10, 21, 25dicopelval2 34548 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  -> 
( <. G ,  (  _I  |`  T ) >.  e.  ( J `  N )  <->  ( G  =  ( (  _I  |`  T ) `  G
)  /\  (  _I  |`  T )  e.  E
) ) )
271, 8, 26syl2anc 656 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( <. G ,  (  _I  |`  T )
>.  e.  ( J `  N )  <->  ( G  =  ( (  _I  |`  T ) `  G
)  /\  (  _I  |`  T )  e.  E
) ) )
2815, 18, 27mpbir2and 908 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  <. G , 
(  _I  |`  T )
>.  e.  ( J `  N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   <.cop 3880   class class class wbr 4289    e. cmpt 4347    _I cid 4627    |` cres 4838   ` cfv 5415   iota_crio 6048  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   lecple 14241   occoc 14242   joincjn 15110   LSSumclsm 16126   Atomscatm 32630   HLchlt 32717   LHypclh 33350   LTrncltrn 33467   trLctrl 33524   TEndoctendo 34118   DVecHcdvh 34445   DIsoBcdib 34505   DIsoCcdic 34539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-riotaBAD 32326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-undef 6788  df-map 7212  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866  df-lines 32867  df-psubsp 32869  df-pmap 32870  df-padd 33162  df-lhyp 33354  df-laut 33355  df-ldil 33470  df-ltrn 33471  df-trl 33525  df-tendo 34121  df-dic 34540
This theorem is referenced by:  cdlemn11b  34575
  Copyright terms: Public domain W3C validator