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Theorem dihpN 32232
Description: The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 31030 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 2442 . 2  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 dihp.n . 2  |-  N  =  ( LSpan `  U )
3 eqid 2442 . 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 32005 . 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 32231 . 2  |-  ( ph  ->  ( I `  P
)  e.  (LSAtoms `  U
) )
11 eqid 2442 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2442 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
1311, 12, 4, 8lhpocnel2 30914 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  (
Atoms `  K )  /\  -.  P ( le `  K ) W ) )
146, 13syl 16 . . . . . . 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 2442 . . . . . . . 8  |-  ( iota_ f  e.  T ( f `
 P )  =  P )  =  (
iota_ f  e.  T
( f `  P
)  =  P )
1815, 11, 12, 4, 16, 17ltrniotaidvalN 31478 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  T
( f `  P
)  =  P )  =  (  _I  |`  B ) )
2019fveq2d 5761 . . . . 5  |-  ( ph  ->  ( S `  ( iota_ f  e.  T ( f `  P )  =  P ) )  =  ( S `  (  _I  |`  B ) ) )
21 dihp.s . . . . . . 7  |-  ( ph  ->  ( S  e.  E  /\  S  =/=  O
) )
2221simpld 447 . . . . . 6  |-  ( ph  ->  S  e.  E )
23 dihp.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
2415, 4, 23tendoid 31668 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
256, 22, 24syl2anc 644 . . . . 5  |-  ( ph  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
2620, 25eqtr2d 2475 . . . 4  |-  ( ph  ->  (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T
( f `  P
)  =  P ) ) )
27 fvex 5771 . . . . . . 7  |-  ( Base `  K )  e.  _V
2815, 27eqeltri 2512 . . . . . 6  |-  B  e. 
_V
29 resiexg 5217 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
3028, 29mp1i 12 . . . . 5  |-  ( ph  ->  (  _I  |`  B )  e.  _V )
31 eqeq1 2448 . . . . . . 7  |-  ( g  =  (  _I  |`  B )  ->  ( g  =  ( s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  <-> 
(  _I  |`  B )  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) ) ) )
3231anbi1d 687 . . . . . 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 5756 . . . . . . . 8  |-  ( s  =  S  ->  (
s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  =  ( S `  ( iota_ f  e.  T ( f `
 P )  =  P ) ) )
3433eqeq2d 2453 . . . . . . 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 2502 . . . . . . 7  |-  ( s  =  S  ->  (
s  e.  E  <->  S  e.  E ) )
3634, 35anbi12d 693 . . . . . 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 4502 . . . . 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 644 . . . 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 890 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  { <. g ,  s >.  |  ( g  =  ( s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  /\  s  e.  E
) } )
40 eqid 2442 . . . . . 6  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
4111, 12, 4, 40, 9dihvalcqat 32135 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  ( Atoms `  K )  /\  -.  P ( le
`  K ) W ) )  ->  (
I `  P )  =  ( ( (
DIsoC `  K ) `  W ) `  P
) )
426, 14, 41syl2anc 644 . . . 4  |-  ( ph  ->  ( I `  P
)  =  ( ( ( DIsoC `  K ) `  W ) `  P
) )
4311, 12, 4, 8, 16, 23, 40dicval 32072 . . . . 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 644 . . . 4  |-  ( ph  ->  ( ( ( DIsoC `  K ) `  W
) `  P )  =  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) )  /\  s  e.  E ) } )
4542, 44eqtr2d 2475 . . 3  |-  ( ph  ->  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) )  /\  s  e.  E ) }  =  ( I `  P
) )
4639, 45eleqtrd 2518 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  ( I `  P ) )
4721simprd 451 . . 3  |-  ( ph  ->  S  =/=  O )
48 dihp.o . . . . . . . 8  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
4915, 4, 16, 5, 1, 48dvh0g 32007 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
506, 49syl 16 . . . . . 6  |-  ( ph  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
5150eqeq2d 2453 . . . . 5  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  <->  <. (  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >. ) )
5228, 29ax-mp 5 . . . . . . 7  |-  (  _I  |`  B )  e.  _V
53 fvex 5771 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
5416, 53eqeltri 2512 . . . . . . . . 9  |-  T  e. 
_V
5554mptex 5995 . . . . . . . 8  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
5648, 55eqeltri 2512 . . . . . . 7  |-  O  e. 
_V
5752, 56opth2 4467 . . . . . 6  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( (  _I  |`  B )  =  (  _I  |`  B )  /\  S  =  O
) )
5857simprbi 452 . . . . 5  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  ->  S  =  O )
5951, 58syl6bi 221 . . . 4  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  ->  S  =  O ) )
6059necon3d 2645 . . 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 29901 1  |-  ( ph  ->  ( I `  P
)  =  ( N `
 { <. (  _I  |`  B ) ,  S >. } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   _Vcvv 2962   {csn 3838   <.cop 3841   class class class wbr 4237   {copab 4290    e. cmpt 4291    _I cid 4522    |` cres 4909   ` cfv 5483   iota_crio 6571   Basecbs 13500   lecple 13567   occoc 13568   0gc0g 13754   LSpanclspn 16078  LSAtomsclsa 29870   Atomscatm 30159   HLchlt 30246   LHypclh 30879   LTrncltrn 30996   TEndoctendo 31647   DVecHcdvh 31974   DIsoCcdic 32068   DIsoHcdih 32124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-tpos 6508  df-undef 6572  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-0g 13758  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-mnd 14721  df-submnd 14770  df-grp 14843  df-minusg 14844  df-sbg 14845  df-subg 14972  df-cntz 15147  df-lsm 15301  df-cmn 15445  df-abl 15446  df-mgp 15680  df-rng 15694  df-ur 15696  df-oppr 15759  df-dvdsr 15777  df-unit 15778  df-invr 15808  df-dvr 15819  df-drng 15868  df-lmod 15983  df-lss 16040  df-lsp 16079  df-lvec 16206  df-lsatoms 29872  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393  df-lplanes 30394  df-lvols 30395  df-lines 30396  df-psubsp 30398  df-pmap 30399  df-padd 30691  df-lhyp 30883  df-laut 30884  df-ldil 30999  df-ltrn 31000  df-trl 31054  df-tendo 31650  df-edring 31652  df-disoa 31925  df-dvech 31975  df-dib 32035  df-dic 32069  df-dih 32125
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