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

Theorem cdlemn11a 37331
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 994 . . . . 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 36140 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
763ad2ant1 1015 . . . . 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 1028 . . . . 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 36702 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  ->  G  e.  T )
121, 7, 8, 11syl3anc 1226 . . . 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 37141 . . . 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 2462 . 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 36892 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
18173ad2ant1 1015 . 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 6236 . . . . 5  |-  ( iota_ h  e.  T  ( h `
 P )  =  N )  e.  _V
2110, 20eqeltri 2538 . . . 4  |-  G  e. 
_V
22 fvex 5858 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
239, 22eqeltri 2538 . . . . 5  |-  T  e. 
_V
24 resiexg 6709 . . . . 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 37305 . . 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 659 . 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 920 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   <.cop 4022   class class class wbr 4439    |-> cmpt 4497    _I cid 4779    |` cres 4990   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   lecple 14791   occoc 14792   joincjn 15772   LSSumclsm 16853   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222   trLctrl 36280   TEndoctendo 36875   DVecHcdvh 37202   DIsoBcdib 37262   DIsoCcdic 37296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281  df-tendo 36878  df-dic 37297
This theorem is referenced by:  cdlemn11b  37332
  Copyright terms: Public domain W3C validator