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

Theorem dihpN 31585
Description: The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 30383 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 2366 . 2  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 dihp.n . 2  |-  N  =  ( LSpan `  U )
3 eqid 2366 . 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 31358 . 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 31584 . 2  |-  ( ph  ->  ( I `  P
)  e.  (LSAtoms `  U
) )
11 eqid 2366 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2366 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
1311, 12, 4, 8lhpocnel2 30267 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  (
Atoms `  K )  /\  -.  P ( le `  K ) W ) )
146, 13syl 15 . . . . . . 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 2366 . . . . . . . 8  |-  ( iota_ f  e.  T ( f `
 P )  =  P )  =  (
iota_ f  e.  T
( f `  P
)  =  P )
1815, 11, 12, 4, 16, 17ltrniotaidvalN 30831 . . . . . . 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 642 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  T
( f `  P
)  =  P )  =  (  _I  |`  B ) )
2019fveq2d 5636 . . . . 5  |-  ( ph  ->  ( S `  ( iota_ f  e.  T ( f `  P )  =  P ) )  =  ( S `  (  _I  |`  B ) ) )
21 dihp.s . . . . . . 7  |-  ( ph  ->  ( S  e.  E  /\  S  =/=  O
) )
2221simpld 445 . . . . . 6  |-  ( ph  ->  S  e.  E )
23 dihp.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
2415, 4, 23tendoid 31021 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
256, 22, 24syl2anc 642 . . . . 5  |-  ( ph  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
2620, 25eqtr2d 2399 . . . 4  |-  ( ph  ->  (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T
( f `  P
)  =  P ) ) )
27 fvex 5646 . . . . . . 7  |-  ( Base `  K )  e.  _V
2815, 27eqeltri 2436 . . . . . 6  |-  B  e. 
_V
29 resiexg 5100 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
3028, 29mp1i 11 . . . . 5  |-  ( ph  ->  (  _I  |`  B )  e.  _V )
31 eqeq1 2372 . . . . . . 7  |-  ( g  =  (  _I  |`  B )  ->  ( g  =  ( s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  <-> 
(  _I  |`  B )  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) ) ) )
3231anbi1d 685 . . . . . 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 5631 . . . . . . . 8  |-  ( s  =  S  ->  (
s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  =  ( S `  ( iota_ f  e.  T ( f `
 P )  =  P ) ) )
3433eqeq2d 2377 . . . . . . 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 2426 . . . . . . 7  |-  ( s  =  S  ->  (
s  e.  E  <->  S  e.  E ) )
3634, 35anbi12d 691 . . . . . 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 4386 . . . . 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 642 . . . 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 888 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  { <. g ,  s >.  |  ( g  =  ( s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  /\  s  e.  E
) } )
40 eqid 2366 . . . . . 6  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
4111, 12, 4, 40, 9dihvalcqat 31488 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  ( Atoms `  K )  /\  -.  P ( le
`  K ) W ) )  ->  (
I `  P )  =  ( ( (
DIsoC `  K ) `  W ) `  P
) )
426, 14, 41syl2anc 642 . . . 4  |-  ( ph  ->  ( I `  P
)  =  ( ( ( DIsoC `  K ) `  W ) `  P
) )
4311, 12, 4, 8, 16, 23, 40dicval 31425 . . . . 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 642 . . . 4  |-  ( ph  ->  ( ( ( DIsoC `  K ) `  W
) `  P )  =  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) )  /\  s  e.  E ) } )
4542, 44eqtr2d 2399 . . 3  |-  ( ph  ->  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) )  /\  s  e.  E ) }  =  ( I `  P
) )
4639, 45eleqtrd 2442 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  ( I `  P ) )
4721simprd 449 . . 3  |-  ( ph  ->  S  =/=  O )
48 dihp.o . . . . . . . 8  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
4915, 4, 16, 5, 1, 48dvh0g 31360 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
506, 49syl 15 . . . . . 6  |-  ( ph  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
5150eqeq2d 2377 . . . . 5  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  <->  <. (  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >. ) )
5228, 29ax-mp 8 . . . . . . 7  |-  (  _I  |`  B )  e.  _V
53 fvex 5646 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
5416, 53eqeltri 2436 . . . . . . . . 9  |-  T  e. 
_V
5554mptex 5866 . . . . . . . 8  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
5648, 55eqeltri 2436 . . . . . . 7  |-  O  e. 
_V
5752, 56opth2 4351 . . . . . 6  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( (  _I  |`  B )  =  (  _I  |`  B )  /\  S  =  O
) )
5857simprbi 450 . . . . 5  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  ->  S  =  O )
5951, 58syl6bi 219 . . . 4  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  ->  S  =  O ) )
6059necon3d 2567 . . 3  |-  ( ph  ->  ( S  =/=  O  -> 
<. (  _I  |`  B ) ,  S >.  =/=  ( 0g `  U ) ) )
6147, 60mpd 14 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  =/=  ( 0g `  U ) )
621, 2, 3, 7, 10, 46, 61lsatel 29254 1  |-  ( ph  ->  ( I `  P
)  =  ( N `
 { <. (  _I  |`  B ) ,  S >. } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   _Vcvv 2873   {csn 3729   <.cop 3732   class class class wbr 4125   {copab 4178    e. cmpt 4179    _I cid 4407    |` cres 4794   ` cfv 5358   iota_crio 6439   Basecbs 13356   lecple 13423   occoc 13424   0gc0g 13610   LSpanclspn 15938  LSAtomsclsa 29223   Atomscatm 29512   HLchlt 29599   LHypclh 30232   LTrncltrn 30349   TEndoctendo 31000   DVecHcdvh 31327   DIsoCcdic 31421   DIsoHcdih 31477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-fal 1325  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-tpos 6376  df-undef 6440  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-0g 13614  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-mnd 14577  df-submnd 14626  df-grp 14699  df-minusg 14700  df-sbg 14701  df-subg 14828  df-cntz 15003  df-lsm 15157  df-cmn 15301  df-abl 15302  df-mgp 15536  df-rng 15550  df-ur 15552  df-oppr 15615  df-dvdsr 15633  df-unit 15634  df-invr 15664  df-dvr 15675  df-drng 15724  df-lmod 15839  df-lss 15900  df-lsp 15939  df-lvec 16066  df-lsatoms 29225  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-llines 29746  df-lplanes 29747  df-lvols 29748  df-lines 29749  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-lhyp 30236  df-laut 30237  df-ldil 30352  df-ltrn 30353  df-trl 30407  df-tendo 31003  df-edring 31005  df-disoa 31278  df-dvech 31328  df-dib 31388  df-dic 31422  df-dih 31478
  Copyright terms: Public domain W3C validator