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

Theorem dihpN 34616
Description: The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 33412 and a nonzero member of the endomorphism ring). In particular,  S can be replaced with the ring unit  (  _I  |`  T ). (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihp.b  |-  B  =  ( Base `  K
)
dihp.h  |-  H  =  ( LHyp `  K
)
dihp.p  |-  P  =  ( ( oc `  K ) `  W
)
dihp.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihp.e  |-  E  =  ( ( TEndo `  K
) `  W )
dihp.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
dihp.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihp.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihp.n  |-  N  =  ( LSpan `  U )
dihp.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dihp.s  |-  ( ph  ->  ( S  e.  E  /\  S  =/=  O
) )
Assertion
Ref Expression
dihpN  |-  ( ph  ->  ( I `  P
)  =  ( N `
 { <. (  _I  |`  B ) ,  S >. } ) )
Distinct variable groups:    B, f    f, H    f, K    P, f    T, f    f, W
Allowed substitution hints:    ph( f)    S( f)    U( f)    E( f)    I( f)    N( f)    O( f)

Proof of Theorem dihpN
Dummy variables  g 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . 2  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 dihp.n . 2  |-  N  =  ( LSpan `  U )
3 eqid 2429 . 2  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
4 dihp.h . . 3  |-  H  =  ( LHyp `  K
)
5 dihp.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
6 dihp.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
74, 5, 6dvhlvec 34389 . 2  |-  ( ph  ->  U  e.  LVec )
8 dihp.p . . 3  |-  P  =  ( ( oc `  K ) `  W
)
9 dihp.i . . 3  |-  I  =  ( ( DIsoH `  K
) `  W )
104, 8, 9, 5, 3, 6dihat 34615 . 2  |-  ( ph  ->  ( I `  P
)  e.  (LSAtoms `  U
) )
11 eqid 2429 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2429 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
1311, 12, 4, 8lhpocnel2 33296 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  (
Atoms `  K )  /\  -.  P ( le `  K ) W ) )
146, 13syl 17 . . . . . . 7  |-  ( ph  ->  ( P  e.  (
Atoms `  K )  /\  -.  P ( le `  K ) W ) )
15 dihp.b . . . . . . . 8  |-  B  =  ( Base `  K
)
16 dihp.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
17 eqid 2429 . . . . . . . 8  |-  ( iota_ f  e.  T  ( f `
 P )  =  P )  =  (
iota_ f  e.  T  ( f `  P
)  =  P )
1815, 11, 12, 4, 16, 17ltrniotaidvalN 33862 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  ( Atoms `  K )  /\  -.  P ( le
`  K ) W ) )  ->  ( iota_ f  e.  T  ( f `  P )  =  P )  =  (  _I  |`  B ) )
196, 14, 18syl2anc 665 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  T  ( f `  P
)  =  P )  =  (  _I  |`  B ) )
2019fveq2d 5885 . . . . 5  |-  ( ph  ->  ( S `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  =  ( S `  (  _I  |`  B ) ) )
21 dihp.s . . . . . . 7  |-  ( ph  ->  ( S  e.  E  /\  S  =/=  O
) )
2221simpld 460 . . . . . 6  |-  ( ph  ->  S  e.  E )
23 dihp.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
2415, 4, 23tendoid 34052 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
256, 22, 24syl2anc 665 . . . . 5  |-  ( ph  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
2620, 25eqtr2d 2471 . . . 4  |-  ( ph  ->  (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T  ( f `  P
)  =  P ) ) )
27 fvex 5891 . . . . . . 7  |-  ( Base `  K )  e.  _V
2815, 27eqeltri 2513 . . . . . 6  |-  B  e. 
_V
29 resiexg 6743 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
3028, 29mp1i 13 . . . . 5  |-  ( ph  ->  (  _I  |`  B )  e.  _V )
31 eqeq1 2433 . . . . . . 7  |-  ( g  =  (  _I  |`  B )  ->  ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  <-> 
(  _I  |`  B )  =  ( s `  ( iota_ f  e.  T  ( f `  P
)  =  P ) ) ) )
3231anbi1d 709 . . . . . 6  |-  ( g  =  (  _I  |`  B )  ->  ( ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P
)  =  P ) )  /\  s  e.  E )  <->  ( (  _I  |`  B )  =  ( s `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  s  e.  E
) ) )
33 fveq1 5880 . . . . . . . 8  |-  ( s  =  S  ->  (
s `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  =  ( S `  ( iota_ f  e.  T  ( f `
 P )  =  P ) ) )
3433eqeq2d 2443 . . . . . . 7  |-  ( s  =  S  ->  (
(  _I  |`  B )  =  ( s `  ( iota_ f  e.  T  ( f `  P
)  =  P ) )  <->  (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T  ( f `  P
)  =  P ) ) ) )
35 eleq1 2501 . . . . . . 7  |-  ( s  =  S  ->  (
s  e.  E  <->  S  e.  E ) )
3634, 35anbi12d 715 . . . . . 6  |-  ( s  =  S  ->  (
( (  _I  |`  B )  =  ( s `  ( iota_ f  e.  T  ( f `  P
)  =  P ) )  /\  s  e.  E )  <->  ( (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  S  e.  E
) ) )
3732, 36opelopabg 4739 . . . . 5  |-  ( ( (  _I  |`  B )  e.  _V  /\  S  e.  E )  ->  ( <. (  _I  |`  B ) ,  S >.  e.  { <. g ,  s >.  |  ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  s  e.  E
) }  <->  ( (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  S  e.  E
) ) )
3830, 22, 37syl2anc 665 . . . 4  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  e.  { <. g ,  s >.  |  ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  s  e.  E
) }  <->  ( (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  S  e.  E
) ) )
3926, 22, 38mpbir2and 930 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  { <. g ,  s >.  |  ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  s  e.  E
) } )
40 eqid 2429 . . . . . 6  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
4111, 12, 4, 40, 9dihvalcqat 34519 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  ( Atoms `  K )  /\  -.  P ( le
`  K ) W ) )  ->  (
I `  P )  =  ( ( (
DIsoC `  K ) `  W ) `  P
) )
426, 14, 41syl2anc 665 . . . 4  |-  ( ph  ->  ( I `  P
)  =  ( ( ( DIsoC `  K ) `  W ) `  P
) )
4311, 12, 4, 8, 16, 23, 40dicval 34456 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  ( Atoms `  K )  /\  -.  P ( le
`  K ) W ) )  ->  (
( ( DIsoC `  K
) `  W ) `  P )  =  { <. g ,  s >.  |  ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P )  =  P ) )  /\  s  e.  E
) } )
446, 14, 43syl2anc 665 . . . 4  |-  ( ph  ->  ( ( ( DIsoC `  K ) `  W
) `  P )  =  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P
)  =  P ) )  /\  s  e.  E ) } )
4542, 44eqtr2d 2471 . . 3  |-  ( ph  ->  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T  ( f `  P
)  =  P ) )  /\  s  e.  E ) }  =  ( I `  P
) )
4639, 45eleqtrd 2519 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  ( I `  P ) )
4721simprd 464 . . 3  |-  ( ph  ->  S  =/=  O )
48 dihp.o . . . . . . . 8  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
4915, 4, 16, 5, 1, 48dvh0g 34391 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
506, 49syl 17 . . . . . 6  |-  ( ph  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
5150eqeq2d 2443 . . . . 5  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  <->  <. (  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >. ) )
5228, 29ax-mp 5 . . . . . . 7  |-  (  _I  |`  B )  e.  _V
53 fvex 5891 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
5416, 53eqeltri 2513 . . . . . . . . 9  |-  T  e. 
_V
5554mptex 6151 . . . . . . . 8  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
5648, 55eqeltri 2513 . . . . . . 7  |-  O  e. 
_V
5752, 56opth2 4700 . . . . . 6  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( (  _I  |`  B )  =  (  _I  |`  B )  /\  S  =  O
) )
5857simprbi 465 . . . . 5  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  ->  S  =  O )
5951, 58syl6bi 231 . . . 4  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  ->  S  =  O ) )
6059necon3d 2655 . . 3  |-  ( ph  ->  ( S  =/=  O  -> 
<. (  _I  |`  B ) ,  S >.  =/=  ( 0g `  U ) ) )
6147, 60mpd 15 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  =/=  ( 0g `  U ) )
621, 2, 3, 7, 10, 46, 61lsatel 32283 1  |-  ( ph  ->  ( I `  P
)  =  ( N `
 { <. (  _I  |`  B ) ,  S >. } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087   {csn 4002   <.cop 4008   class class class wbr 4426   {copab 4483    |-> cmpt 4484    _I cid 4764    |` cres 4856   ` cfv 5601   iota_crio 6266   Basecbs 15084   lecple 15159   occoc 15160   0gc0g 15297   LSpanclspn 18129  LSAtomsclsa 32252   Atomscatm 32541   HLchlt 32628   LHypclh 33261   LTrncltrn 33378   TEndoctendo 34031   DVecHcdvh 34358   DIsoCcdic 34452   DIsoHcdih 34508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32237
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-cntz 16922  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lvec 18261  df-lsatoms 32254  df-oposet 32454  df-ol 32456  df-oml 32457  df-covers 32544  df-ats 32545  df-atl 32576  df-cvlat 32600  df-hlat 32629  df-llines 32775  df-lplanes 32776  df-lvols 32777  df-lines 32778  df-psubsp 32780  df-pmap 32781  df-padd 33073  df-lhyp 33265  df-laut 33266  df-ldil 33381  df-ltrn 33382  df-trl 33437  df-tendo 34034  df-edring 34036  df-disoa 34309  df-dvech 34359  df-dib 34419  df-dic 34453  df-dih 34509
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator