MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrinvcl Structured version   Unicode version

Theorem dchrinvcl 21037
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 21024 . . . . 5  |-  ( X  e.  D  ->  N  e.  NN )
96, 8syl 16 . . . 4  |-  ( ph  ->  N  e.  NN )
10 fveq2 5728 . . . . 5  |-  ( k  =  x  ->  ( X `  k )  =  ( X `  x ) )
1110oveq2d 6097 . . . 4  |-  ( k  =  x  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  x
) ) )
12 fveq2 5728 . . . . 5  |-  ( k  =  y  ->  ( X `  k )  =  ( X `  y ) )
1312oveq2d 6097 . . . 4  |-  ( k  =  y  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  y
) ) )
14 fveq2 5728 . . . . 5  |-  ( k  =  ( x ( .r `  Z ) y )  ->  ( X `  k )  =  ( X `  ( x ( .r
`  Z ) y ) ) )
1514oveq2d 6097 . . . 4  |-  ( k  =  ( x ( .r `  Z ) y )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  (
x ( .r `  Z ) y ) ) ) )
16 fveq2 5728 . . . . 5  |-  ( k  =  ( 1r `  Z )  ->  ( X `  k )  =  ( X `  ( 1r `  Z ) ) )
1716oveq2d 6097 . . . 4  |-  ( k  =  ( 1r `  Z )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  ( 1r `  Z ) ) ) )
182, 3, 7, 4, 6dchrf 21026 . . . . . 6  |-  ( ph  ->  X : B --> CC )
194, 5unitss 15765 . . . . . . 7  |-  U  C_  B
2019sseli 3344 . . . . . 6  |-  ( k  e.  U  ->  k  e.  B )
21 ffvelrn 5868 . . . . . 6  |-  ( ( X : B --> CC  /\  k  e.  B )  ->  ( X `  k
)  e.  CC )
2218, 20, 21syl2an 464 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
23 simpr 448 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  U )
246adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  X  e.  D )
2520adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  B )
262, 3, 7, 4, 5, 24, 25dchrn0 21034 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
2723, 26mpbird 224 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
2822, 27reccld 9783 . . . 4  |-  ( (
ph  /\  k  e.  U )  ->  (
1  /  ( X `
 k ) )  e.  CC )
29 1t1e1 10126 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
3029eqcomi 2440 . . . . . . 7  |-  1  =  ( 1  x.  1 )
3130a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  =  ( 1  x.  1 ) )
322, 3, 7dchrmhm 21025 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
336adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  D )
3432, 33sseldi 3346 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
35 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  U )
3619, 35sseldi 3346 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  B )
37 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  U )
3819, 37sseldi 3346 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  B )
39 eqid 2436 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
4039, 4mgpbas 15654 . . . . . . . 8  |-  B  =  ( Base `  (mulGrp `  Z ) )
41 eqid 2436 . . . . . . . . 9  |-  ( .r
`  Z )  =  ( .r `  Z
)
4239, 41mgpplusg 15652 . . . . . . . 8  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
43 eqid 2436 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
44 cnfldmul 16709 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
4543, 44mgpplusg 15652 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4640, 42, 45mhmlin 14745 . . . . . . 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 1184 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  (
x ( .r `  Z ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) ) )
4831, 47oveq12d 6099 . . . . 5  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  x.  1 )  /  ( ( X `
 x )  x.  ( X `  y
) ) ) )
49 ax-1cn 9048 . . . . . . 7  |-  1  e.  CC
5049a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  e.  CC )
5118adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X : B --> CC )
5251, 36ffvelrnd 5871 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  e.  CC )
5351, 38ffvelrnd 5871 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  e.  CC )
542, 3, 7, 4, 5, 33, 36dchrn0 21034 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  x )  =/=  0  <->  x  e.  U ) )
5535, 54mpbird 224 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  =/=  0 )
562, 3, 7, 4, 5, 33, 38dchrn0 21034 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  y )  =/=  0  <->  y  e.  U ) )
5737, 56mpbird 224 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  =/=  0 )
5850, 52, 50, 53, 55, 57divmuldivd 9831 . . . . 5  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( 1  / 
( X `  x
) )  x.  (
1  /  ( X `
 y ) ) )  =  ( ( 1  x.  1 )  /  ( ( X `
 x )  x.  ( X `  y
) ) ) )
5948, 58eqtr4d 2471 . . . 4  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  /  ( X `
 x ) )  x.  ( 1  / 
( X `  y
) ) ) )
6032, 6sseldi 3346 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
61 eqid 2436 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
6239, 61rngidval 15666 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
63 cnfld1 16726 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
6443, 63rngidval 15666 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
6562, 64mhm0 14746 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
6660, 65syl 16 . . . . . 6  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
6766oveq2d 6097 . . . . 5  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  ( 1  / 
1 ) )
6849div1i 9742 . . . . 5  |-  ( 1  /  1 )  =  1
6967, 68syl6eq 2484 . . . 4  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  1 )
702, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 59, 69dchrelbasd 21023 . . 3  |-  ( ph  ->  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )  e.  D )
711, 70syl5eqel 2520 . 2  |-  ( ph  ->  K  e.  D )
72 dchrmulid2.t . . . 4  |-  .x.  =  ( +g  `  G )
732, 3, 7, 72, 71, 6dchrmul 21032 . . 3  |-  ( ph  ->  ( K  .x.  X
)  =  ( K  o F  x.  X
) )
74 fvex 5742 . . . . . . 7  |-  ( Base `  Z )  e.  _V
754, 74eqeltri 2506 . . . . . 6  |-  B  e. 
_V
7675a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
77 ovex 6106 . . . . . . 7  |-  ( 1  /  ( X `  k ) )  e. 
_V
78 c0ex 9085 . . . . . . 7  |-  0  e.  _V
7977, 78ifex 3797 . . . . . 6  |-  if ( k  e.  U , 
( 1  /  ( X `  k )
) ,  0 )  e.  _V
8079a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  e.  _V )
8118ffvelrnda 5870 . . . . 5  |-  ( (
ph  /\  k  e.  B )  ->  ( X `  k )  e.  CC )
821a1i 11 . . . . 5  |-  ( ph  ->  K  =  ( k  e.  B  |->  if ( k  e.  U , 
( 1  /  ( X `  k )
) ,  0 ) ) )
8318feqmptd 5779 . . . . 5  |-  ( ph  ->  X  =  ( k  e.  B  |->  ( X `
 k ) ) )
8476, 80, 81, 82, 83offval2 6322 . . . 4  |-  ( ph  ->  ( K  o F  x.  X )  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) ) ) )
85 oveq1 6088 . . . . . . . 8  |-  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  =  ( 1  /  ( X `  k ) )  -> 
( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) )  =  ( ( 1  /  ( X `  k )
)  x.  ( X `
 k ) ) )
86 oveq1 6088 . . . . . . . 8  |-  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  =  0  -> 
( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) )  =  ( 0  x.  ( X `
 k ) ) )
8785, 86ifsb 3748 . . . . . . 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
) ) )
8881adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
896adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  X  e.  D )
90 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  k  e.  B )
912, 3, 7, 4, 5, 89, 90dchrn0 21034 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  B )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
9291biimpar 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
9388, 92recid2d 9786 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  (
( 1  /  ( X `  k )
)  x.  ( X `
 k ) )  =  1 )
9493ifeq1da 3764 . . . . . . . 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 ) ) ) )
9581mul02d 9264 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  B )  ->  (
0  x.  ( X `
 k ) )  =  0 )
9695ifeq2d 3754 . . . . . . . 8  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  1 ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  0 ) )
9794, 96eqtrd 2468 . . . . . . 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 ) )
9887, 97syl5eq 2480 . . . . . 6  |-  ( (
ph  /\  k  e.  B )  ->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) )  =  if ( k  e.  U ,  1 ,  0 ) )
9998mpteq2dva 4295 . . . . 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 ) ) )
100 dchr1cl.o . . . . 5  |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )
10199, 100syl6reqr 2487 . . . 4  |-  ( ph  ->  .1.  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) ) ) )
10284, 101eqtr4d 2471 . . 3  |-  ( ph  ->  ( K  o F  x.  X )  =  .1.  )
10373, 102eqtrd 2468 . 2  |-  ( ph  ->  ( K  .x.  X
)  =  .1.  )
10471, 103jca 519 1  |-  ( ph  ->  ( K  e.  D  /\  ( K  .x.  X
)  =  .1.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   ifcif 3739    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995    / cdiv 9677   NNcn 10000   Basecbs 13469   +g cplusg 13529   .rcmulr 13530   MndHom cmhm 14736  mulGrpcmgp 15648   1rcur 15662  Unitcui 15744  ℂfldccnfld 16703  ℤ/nczn 16781  DChrcdchr 21016
This theorem is referenced by:  dchrabl  21038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-imas 13734  df-divs 13735  df-mnd 14690  df-mhm 14738  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-nsg 14942  df-eqg 14943  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-sra 16244  df-rgmod 16245  df-lidl 16246  df-rsp 16247  df-2idl 16303  df-cnfld 16704  df-zn 16785  df-dchr 21017
  Copyright terms: Public domain W3C validator