Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dih1dimb2 Structured version   Unicode version

Theorem dih1dimb2 31940
Description: Isomorphism H at an atom under  W. (Contributed by NM, 27-Apr-2014.)
Hypotheses
Ref Expression
dih1dimb2.b  |-  B  =  ( Base `  K
)
dih1dimb2.l  |-  .<_  =  ( le `  K )
dih1dimb2.a  |-  A  =  ( Atoms `  K )
dih1dimb2.h  |-  H  =  ( LHyp `  K
)
dih1dimb2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dih1dimb2.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
dih1dimb2.u  |-  U  =  ( ( DVecH `  K
) `  W )
dih1dimb2.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dih1dimb2.n  |-  N  =  ( LSpan `  U )
Assertion
Ref Expression
dih1dimb2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
Distinct variable groups:    .<_ , f    A, f    B, h    f, H   
f, h, K    Q, f    T, f, h    f, W, h
Allowed substitution hints:    A( h)    B( f)    Q( h)    U( f, h)    H( h)    I( f, h)   
.<_ ( h)    N( f, h)    O( f, h)

Proof of Theorem dih1dimb2
StepHypRef Expression
1 dih1dimb2.l . . 3  |-  .<_  =  ( le `  K )
2 dih1dimb2.a . . 3  |-  A  =  ( Atoms `  K )
3 dih1dimb2.h . . 3  |-  H  =  ( LHyp `  K
)
4 dih1dimb2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
5 eqid 2435 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
61, 2, 3, 4, 5cdlemf 31261 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  E. f  e.  T  ( (
( trL `  K
) `  W ) `  f )  =  Q )
7 simp3 959 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
( ( trL `  K
) `  W ) `  f )  =  Q )
8 simp1rl 1022 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  Q  e.  A )
97, 8eqeltrd 2509 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
( ( trL `  K
) `  W ) `  f )  e.  A
)
10 simp1l 981 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simp2 958 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  f  e.  T )
12 dih1dimb2.b . . . . . . . 8  |-  B  =  ( Base `  K
)
1312, 2, 3, 4, 5trlnidatb 30875 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( f  =/=  (  _I  |`  B )  <-> 
( ( ( trL `  K ) `  W
) `  f )  e.  A ) )
1410, 11, 13syl2anc 643 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
f  =/=  (  _I  |`  B )  <->  ( (
( trL `  K
) `  W ) `  f )  e.  A
) )
159, 14mpbird 224 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  f  =/=  (  _I  |`  B ) )
167fveq2d 5724 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
I `  ( (
( trL `  K
) `  W ) `  f ) )  =  ( I `  Q
) )
17 dih1dimb2.o . . . . . . . 8  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
18 dih1dimb2.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
19 dih1dimb2.i . . . . . . . 8  |-  I  =  ( ( DIsoH `  K
) `  W )
20 dih1dimb2.n . . . . . . . 8  |-  N  =  ( LSpan `  U )
2112, 3, 4, 5, 17, 18, 19, 20dih1dimb 31939 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( I `  ( ( ( trL `  K ) `  W
) `  f )
)  =  ( N `
 { <. f ,  O >. } ) )
2210, 11, 21syl2anc 643 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
I `  ( (
( trL `  K
) `  W ) `  f ) )  =  ( N `  { <. f ,  O >. } ) )
2316, 22eqtr3d 2469 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
I `  Q )  =  ( N `  { <. f ,  O >. } ) )
2415, 23jca 519 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
f  =/=  (  _I  |`  B )  /\  (
I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
25243expia 1155 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T
)  ->  ( (
( ( trL `  K
) `  W ) `  f )  =  Q  ->  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) ) )
2625reximdva 2810 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  ( E. f  e.  T  ( ( ( trL `  K ) `  W
) `  f )  =  Q  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) ) )
276, 26mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {csn 3806   <.cop 3809   class class class wbr 4204    e. cmpt 4258    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13459   lecple 13526   LSpanclspn 16037   Atomscatm 29962   HLchlt 30049   LHypclh 30682   LTrncltrn 30799   trLctrl 30856   DVecHcdvh 31777   DIsoHcdih 31927
This theorem is referenced by:  dihatlat  32033  dihatexv  32037
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-fal 1329  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-tpos 6471  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-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-0g 13717  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-mnd 14680  df-grp 14802  df-minusg 14803  df-sbg 14804  df-mgp 15639  df-rng 15653  df-ur 15655  df-oppr 15718  df-dvdsr 15736  df-unit 15737  df-invr 15767  df-dvr 15778  df-drng 15827  df-lmod 15942  df-lss 15999  df-lsp 16038  df-lvec 16165  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-disoa 31728  df-dvech 31778  df-dib 31838  df-dih 31928
  Copyright terms: Public domain W3C validator