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Theorem dihpN 31831
Description: The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 30629 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 2412 . 2  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 dihp.n . 2  |-  N  =  ( LSpan `  U )
3 eqid 2412 . 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 31604 . 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 31830 . 2  |-  ( ph  ->  ( I `  P
)  e.  (LSAtoms `  U
) )
11 eqid 2412 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2412 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
1311, 12, 4, 8lhpocnel2 30513 . . . . . . . 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 2412 . . . . . . . 8  |-  ( iota_ f  e.  T ( f `
 P )  =  P )  =  (
iota_ f  e.  T
( f `  P
)  =  P )
1815, 11, 12, 4, 16, 17ltrniotaidvalN 31077 . . . . . . 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 643 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  T
( f `  P
)  =  P )  =  (  _I  |`  B ) )
2019fveq2d 5699 . . . . 5  |-  ( ph  ->  ( S `  ( iota_ f  e.  T ( f `  P )  =  P ) )  =  ( S `  (  _I  |`  B ) ) )
21 dihp.s . . . . . . 7  |-  ( ph  ->  ( S  e.  E  /\  S  =/=  O
) )
2221simpld 446 . . . . . 6  |-  ( ph  ->  S  e.  E )
23 dihp.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
2415, 4, 23tendoid 31267 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
256, 22, 24syl2anc 643 . . . . 5  |-  ( ph  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
2620, 25eqtr2d 2445 . . . 4  |-  ( ph  ->  (  _I  |`  B )  =  ( S `  ( iota_ f  e.  T
( f `  P
)  =  P ) ) )
27 fvex 5709 . . . . . . 7  |-  ( Base `  K )  e.  _V
2815, 27eqeltri 2482 . . . . . 6  |-  B  e. 
_V
29 resiexg 5155 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
3028, 29mp1i 12 . . . . 5  |-  ( ph  ->  (  _I  |`  B )  e.  _V )
31 eqeq1 2418 . . . . . . 7  |-  ( g  =  (  _I  |`  B )  ->  ( g  =  ( s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  <-> 
(  _I  |`  B )  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) ) ) )
3231anbi1d 686 . . . . . 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 5694 . . . . . . . 8  |-  ( s  =  S  ->  (
s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  =  ( S `  ( iota_ f  e.  T ( f `
 P )  =  P ) ) )
3433eqeq2d 2423 . . . . . . 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 2472 . . . . . . 7  |-  ( s  =  S  ->  (
s  e.  E  <->  S  e.  E ) )
3634, 35anbi12d 692 . . . . . 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 4441 . . . . 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 643 . . . 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 889 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  { <. g ,  s >.  |  ( g  =  ( s `  ( iota_ f  e.  T ( f `  P )  =  P ) )  /\  s  e.  E
) } )
40 eqid 2412 . . . . . 6  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
4111, 12, 4, 40, 9dihvalcqat 31734 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  ( Atoms `  K )  /\  -.  P ( le
`  K ) W ) )  ->  (
I `  P )  =  ( ( (
DIsoC `  K ) `  W ) `  P
) )
426, 14, 41syl2anc 643 . . . 4  |-  ( ph  ->  ( I `  P
)  =  ( ( ( DIsoC `  K ) `  W ) `  P
) )
4311, 12, 4, 8, 16, 23, 40dicval 31671 . . . . 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 643 . . . 4  |-  ( ph  ->  ( ( ( DIsoC `  K ) `  W
) `  P )  =  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) )  /\  s  e.  E ) } )
4542, 44eqtr2d 2445 . . 3  |-  ( ph  ->  { <. g ,  s
>.  |  ( g  =  ( s `  ( iota_ f  e.  T
( f `  P
)  =  P ) )  /\  s  e.  E ) }  =  ( I `  P
) )
4639, 45eleqtrd 2488 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  S >.  e.  ( I `  P ) )
4721simprd 450 . . 3  |-  ( ph  ->  S  =/=  O )
48 dihp.o . . . . . . . 8  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
4915, 4, 16, 5, 1, 48dvh0g 31606 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
506, 49syl 16 . . . . . 6  |-  ( ph  ->  ( 0g `  U
)  =  <. (  _I  |`  B ) ,  O >. )
5150eqeq2d 2423 . . . . 5  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  <->  <. (  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >. ) )
5228, 29ax-mp 8 . . . . . . 7  |-  (  _I  |`  B )  e.  _V
53 fvex 5709 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
5416, 53eqeltri 2482 . . . . . . . . 9  |-  T  e. 
_V
5554mptex 5933 . . . . . . . 8  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
5648, 55eqeltri 2482 . . . . . . 7  |-  O  e. 
_V
5752, 56opth2 4406 . . . . . 6  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( (  _I  |`  B )  =  (  _I  |`  B )  /\  S  =  O
) )
5857simprbi 451 . . . . 5  |-  ( <.
(  _I  |`  B ) ,  S >.  =  <. (  _I  |`  B ) ,  O >.  ->  S  =  O )
5951, 58syl6bi 220 . . . 4  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  S >.  =  ( 0g `  U )  ->  S  =  O ) )
6059necon3d 2613 . . 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 29500 1  |-  ( ph  ->  ( I `  P
)  =  ( N `
 { <. (  _I  |`  B ) ,  S >. } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924   {csn 3782   <.cop 3785   class class class wbr 4180   {copab 4233    e. cmpt 4234    _I cid 4461    |` cres 4847   ` cfv 5421   iota_crio 6509   Basecbs 13432   lecple 13499   occoc 13500   0gc0g 13686   LSpanclspn 16010  LSAtomsclsa 29469   Atomscatm 29758   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   TEndoctendo 31246   DVecHcdvh 31573   DIsoCcdic 31667   DIsoHcdih 31723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-tpos 6446  df-undef 6510  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-0g 13690  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-cntz 15079  df-lsm 15233  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-dvr 15751  df-drng 15800  df-lmod 15915  df-lss 15972  df-lsp 16011  df-lvec 16138  df-lsatoms 29471  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-tendo 31249  df-edring 31251  df-disoa 31524  df-dvech 31574  df-dib 31634  df-dic 31668  df-dih 31724
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