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Theorem dchrinvcl 24083
Description: Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
dchrmhm.g  |-  G  =  (DChr `  N )
dchrmhm.z  |-  Z  =  (ℤ/n `  N )
dchrmhm.b  |-  D  =  ( Base `  G
)
dchrn0.b  |-  B  =  ( Base `  Z
)
dchrn0.u  |-  U  =  (Unit `  Z )
dchr1cl.o  |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )
dchrmulid2.t  |-  .x.  =  ( +g  `  G )
dchrmulid2.x  |-  ( ph  ->  X  e.  D )
dchrinvcl.n  |-  K  =  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )
Assertion
Ref Expression
dchrinvcl  |-  ( ph  ->  ( K  e.  D  /\  ( K  .x.  X
)  =  .1.  )
)
Distinct variable groups:    B, k    U, k    k, N    ph, k    k, X    k, Z
Allowed substitution hints:    D( k)    .x. ( k)    .1. ( k)    G( k)    K( k)

Proof of Theorem dchrinvcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrinvcl.n . . 3  |-  K  =  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )
2 dchrmhm.g . . . 4  |-  G  =  (DChr `  N )
3 dchrmhm.z . . . 4  |-  Z  =  (ℤ/n `  N )
4 dchrn0.b . . . 4  |-  B  =  ( Base `  Z
)
5 dchrn0.u . . . 4  |-  U  =  (Unit `  Z )
6 dchrmulid2.x . . . . 5  |-  ( ph  ->  X  e.  D )
7 dchrmhm.b . . . . . 6  |-  D  =  ( Base `  G
)
82, 7dchrrcl 24070 . . . . 5  |-  ( X  e.  D  ->  N  e.  NN )
96, 8syl 17 . . . 4  |-  ( ph  ->  N  e.  NN )
10 fveq2 5872 . . . . 5  |-  ( k  =  x  ->  ( X `  k )  =  ( X `  x ) )
1110oveq2d 6312 . . . 4  |-  ( k  =  x  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  x
) ) )
12 fveq2 5872 . . . . 5  |-  ( k  =  y  ->  ( X `  k )  =  ( X `  y ) )
1312oveq2d 6312 . . . 4  |-  ( k  =  y  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  y
) ) )
14 fveq2 5872 . . . . 5  |-  ( k  =  ( x ( .r `  Z ) y )  ->  ( X `  k )  =  ( X `  ( x ( .r
`  Z ) y ) ) )
1514oveq2d 6312 . . . 4  |-  ( k  =  ( x ( .r `  Z ) y )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  (
x ( .r `  Z ) y ) ) ) )
16 fveq2 5872 . . . . 5  |-  ( k  =  ( 1r `  Z )  ->  ( X `  k )  =  ( X `  ( 1r `  Z ) ) )
1716oveq2d 6312 . . . 4  |-  ( k  =  ( 1r `  Z )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  ( 1r `  Z ) ) ) )
182, 3, 7, 4, 6dchrf 24072 . . . . . 6  |-  ( ph  ->  X : B --> CC )
194, 5unitss 17829 . . . . . . 7  |-  U  C_  B
2019sseli 3457 . . . . . 6  |-  ( k  e.  U  ->  k  e.  B )
21 ffvelrn 6026 . . . . . 6  |-  ( ( X : B --> CC  /\  k  e.  B )  ->  ( X `  k
)  e.  CC )
2218, 20, 21syl2an 479 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
23 simpr 462 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  U )
246adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  X  e.  D )
2520adantl 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  B )
262, 3, 7, 4, 5, 24, 25dchrn0 24080 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
2723, 26mpbird 235 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
2822, 27reccld 10365 . . . 4  |-  ( (
ph  /\  k  e.  U )  ->  (
1  /  ( X `
 k ) )  e.  CC )
29 1t1e1 10746 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
3029eqcomi 2433 . . . . . . 7  |-  1  =  ( 1  x.  1 )
3130a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  =  ( 1  x.  1 ) )
322, 3, 7dchrmhm 24071 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
336adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  D )
3432, 33sseldi 3459 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
35 simprl 762 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  U )
3619, 35sseldi 3459 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  B )
37 simprr 764 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  U )
3819, 37sseldi 3459 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  B )
39 eqid 2420 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
4039, 4mgpbas 17670 . . . . . . . 8  |-  B  =  ( Base `  (mulGrp `  Z ) )
41 eqid 2420 . . . . . . . . 9  |-  ( .r
`  Z )  =  ( .r `  Z
)
4239, 41mgpplusg 17668 . . . . . . . 8  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
43 eqid 2420 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
44 cnfldmul 18917 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
4543, 44mgpplusg 17668 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4640, 42, 45mhmlin 16541 . . . . . . 7  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  x  e.  B  /\  y  e.  B
)  ->  ( X `  ( x ( .r
`  Z ) y ) )  =  ( ( X `  x
)  x.  ( X `
 y ) ) )
4734, 36, 38, 46syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  (
x ( .r `  Z ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) ) )
4831, 47oveq12d 6314 . . . . 5  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  x.  1 )  /  ( ( X `
 x )  x.  ( X `  y
) ) ) )
49 1cnd 9648 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  e.  CC )
5018adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X : B --> CC )
5150, 36ffvelrnd 6029 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  e.  CC )
5250, 38ffvelrnd 6029 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  e.  CC )
532, 3, 7, 4, 5, 33, 36dchrn0 24080 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  x )  =/=  0  <->  x  e.  U ) )
5435, 53mpbird 235 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  =/=  0 )
552, 3, 7, 4, 5, 33, 38dchrn0 24080 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  y )  =/=  0  <->  y  e.  U ) )
5637, 55mpbird 235 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  =/=  0 )
5749, 51, 49, 52, 54, 56divmuldivd 10413 . . . . 5  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( 1  / 
( X `  x
) )  x.  (
1  /  ( X `
 y ) ) )  =  ( ( 1  x.  1 )  /  ( ( X `
 x )  x.  ( X `  y
) ) ) )
5848, 57eqtr4d 2464 . . . 4  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  /  ( X `
 x ) )  x.  ( 1  / 
( X `  y
) ) ) )
5932, 6sseldi 3459 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
60 eqid 2420 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
6139, 60ringidval 17678 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
62 cnfld1 18934 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
6343, 62ringidval 17678 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
6461, 63mhm0 16542 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
6559, 64syl 17 . . . . . 6  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
6665oveq2d 6312 . . . . 5  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  ( 1  / 
1 ) )
67 1div1e1 10289 . . . . 5  |-  ( 1  /  1 )  =  1
6866, 67syl6eq 2477 . . . 4  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  1 )
692, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 58, 68dchrelbasd 24069 . . 3  |-  ( ph  ->  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )  e.  D )
701, 69syl5eqel 2512 . 2  |-  ( ph  ->  K  e.  D )
71 dchrmulid2.t . . . 4  |-  .x.  =  ( +g  `  G )
722, 3, 7, 71, 70, 6dchrmul 24078 . . 3  |-  ( ph  ->  ( K  .x.  X
)  =  ( K  oF  x.  X
) )
73 fvex 5882 . . . . . . 7  |-  ( Base `  Z )  e.  _V
744, 73eqeltri 2504 . . . . . 6  |-  B  e. 
_V
7574a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
76 ovex 6324 . . . . . . 7  |-  ( 1  /  ( X `  k ) )  e. 
_V
77 c0ex 9626 . . . . . . 7  |-  0  e.  _V
7876, 77ifex 3974 . . . . . 6  |-  if ( k  e.  U , 
( 1  /  ( X `  k )
) ,  0 )  e.  _V
7978a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  e.  _V )
8018ffvelrnda 6028 . . . . 5  |-  ( (
ph  /\  k  e.  B )  ->  ( X `  k )  e.  CC )
811a1i 11 . . . . 5  |-  ( ph  ->  K  =  ( k  e.  B  |->  if ( k  e.  U , 
( 1  /  ( X `  k )
) ,  0 ) ) )
8218feqmptd 5925 . . . . 5  |-  ( ph  ->  X  =  ( k  e.  B  |->  ( X `
 k ) ) )
8375, 79, 80, 81, 82offval2 6553 . . . 4  |-  ( ph  ->  ( K  oF  x.  X )  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) ) ) )
84 ovif 6378 . . . . . . 7  |-  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) )  =  if ( k  e.  U ,  ( ( 1  /  ( X `  k )
)  x.  ( X `
 k ) ) ,  ( 0  x.  ( X `  k
) ) )
8580adantr 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
866adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  X  e.  D )
87 simpr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  k  e.  B )
882, 3, 7, 4, 5, 86, 87dchrn0 24080 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  B )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
8988biimpar 487 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
9085, 89recid2d 10368 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  (
( 1  /  ( X `  k )
)  x.  ( X `
 k ) )  =  1 )
9190ifeq1da 3936 . . . . . . . 8  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  ( ( 1  /  ( X `  k ) )  x.  ( X `  k
) ) ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  ( 0  x.  ( X `  k ) ) ) )
9280mul02d 9820 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  B )  ->  (
0  x.  ( X `
 k ) )  =  0 )
9392ifeq2d 3925 . . . . . . . 8  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  1 ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  0 ) )
9491, 93eqtrd 2461 . . . . . . 7  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  ( ( 1  /  ( X `  k ) )  x.  ( X `  k
) ) ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  0 ) )
9584, 94syl5eq 2473 . . . . . 6  |-  ( (
ph  /\  k  e.  B )  ->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) )  =  if ( k  e.  U ,  1 ,  0 ) )
9695mpteq2dva 4503 . . . . 5  |-  ( ph  ->  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) ) )  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) ) )
97 dchr1cl.o . . . . 5  |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )
9896, 97syl6reqr 2480 . . . 4  |-  ( ph  ->  .1.  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) ) ) )
9983, 98eqtr4d 2464 . . 3  |-  ( ph  ->  ( K  oF  x.  X )  =  .1.  )
10072, 99eqtrd 2461 . 2  |-  ( ph  ->  ( K  .x.  X
)  =  .1.  )
10170, 100jca 534 1  |-  ( ph  ->  ( K  e.  D  /\  ( K  .x.  X
)  =  .1.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078   ifcif 3906    |-> cmpt 4475   -->wf 5588   ` cfv 5592  (class class class)co 6296    oFcof 6534   CCcc 9526   0cc0 9528   1c1 9529    x. cmul 9533    / cdiv 10258   NNcn 10598   Basecbs 15081   +g cplusg 15150   .rcmulr 15151   MndHom cmhm 16532  mulGrpcmgp 17664   1rcur 17676  Unitcui 17808  ℂfldccnfld 18911  ℤ/nczn 19011  DChrcdchr 24062
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-ec 7364  df-qs 7368  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-plusg 15163  df-mulr 15164  df-starv 15165  df-sca 15166  df-vsca 15167  df-ip 15168  df-tset 15169  df-ple 15170  df-ds 15172  df-unif 15173  df-0g 15300  df-imas 15366  df-qus 15367  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-mhm 16534  df-grp 16625  df-minusg 16626  df-sbg 16627  df-subg 16766  df-nsg 16767  df-eqg 16768  df-cmn 17373  df-abl 17374  df-mgp 17665  df-ur 17677  df-ring 17723  df-cring 17724  df-oppr 17792  df-dvdsr 17810  df-unit 17811  df-invr 17841  df-subrg 17947  df-lmod 18034  df-lss 18097  df-lsp 18136  df-sra 18336  df-rgmod 18337  df-lidl 18338  df-rsp 18339  df-2idl 18397  df-cnfld 18912  df-zring 18980  df-zn 19015  df-dchr 24063
This theorem is referenced by:  dchrabl  24084
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