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Theorem dchrval 21020
 Description: Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
Hypotheses
Ref Expression
dchrval.g DChr
dchrval.z ℤ/n
dchrval.b
dchrval.u Unit
dchrval.n
dchrval.d mulGrp MndHom mulGrpfld
Assertion
Ref Expression
dchrval
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem dchrval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrval.g . 2 DChr
2 df-dchr 21019 . . . 4 DChr ℤ/n mulGrp MndHom mulGrpfld Unit
32a1i 11 . . 3 DChr ℤ/n mulGrp MndHom mulGrpfld Unit
4 fvex 5744 . . . . 5 ℤ/n
54a1i 11 . . . 4 ℤ/n
6 ovex 6108 . . . . . . 7 mulGrp MndHom mulGrpfld
76rabex 4356 . . . . . 6 mulGrp MndHom mulGrpfld Unit
87a1i 11 . . . . 5 ℤ/n mulGrp MndHom mulGrpfld Unit
9 dchrval.d . . . . . . . . . . 11 mulGrp MndHom mulGrpfld
109ad2antrr 708 . . . . . . . . . 10 ℤ/n mulGrp MndHom mulGrpfld
11 simpr 449 . . . . . . . . . . . . . . . . 17
1211fveq2d 5734 . . . . . . . . . . . . . . . 16 ℤ/n ℤ/n
13 dchrval.z . . . . . . . . . . . . . . . 16 ℤ/n
1412, 13syl6reqr 2489 . . . . . . . . . . . . . . 15 ℤ/n
1514eqeq2d 2449 . . . . . . . . . . . . . 14 ℤ/n
1615biimpar 473 . . . . . . . . . . . . 13 ℤ/n
1716fveq2d 5734 . . . . . . . . . . . 12 ℤ/n mulGrp mulGrp
1817oveq1d 6098 . . . . . . . . . . 11 ℤ/n mulGrp MndHom mulGrpfld mulGrp MndHom mulGrpfld
1916fveq2d 5734 . . . . . . . . . . . . . . 15 ℤ/n
20 dchrval.b . . . . . . . . . . . . . . 15
2119, 20syl6eqr 2488 . . . . . . . . . . . . . 14 ℤ/n
2216fveq2d 5734 . . . . . . . . . . . . . . 15 ℤ/n Unit Unit
23 dchrval.u . . . . . . . . . . . . . . 15 Unit
2422, 23syl6eqr 2488 . . . . . . . . . . . . . 14 ℤ/n Unit
2521, 24difeq12d 3468 . . . . . . . . . . . . 13 ℤ/n Unit
2625xpeq1d 4903 . . . . . . . . . . . 12 ℤ/n Unit
2726sseq1d 3377 . . . . . . . . . . 11 ℤ/n Unit
2818, 27rabeqbidv 2953 . . . . . . . . . 10 ℤ/n mulGrp MndHom mulGrpfld Unit mulGrp MndHom mulGrpfld
2910, 28eqtr4d 2473 . . . . . . . . 9 ℤ/n mulGrp MndHom mulGrpfld Unit
3029eqeq2d 2449 . . . . . . . 8 ℤ/n mulGrp MndHom mulGrpfld Unit
3130biimpar 473 . . . . . . 7 ℤ/n mulGrp MndHom mulGrpfld Unit
3231opeq2d 3993 . . . . . 6 ℤ/n mulGrp MndHom mulGrpfld Unit
3331, 31xpeq12d 4905 . . . . . . . 8 ℤ/n mulGrp MndHom mulGrpfld Unit
3433reseq2d 5148 . . . . . . 7 ℤ/n mulGrp MndHom mulGrpfld Unit
3534opeq2d 3993 . . . . . 6 ℤ/n mulGrp MndHom mulGrpfld Unit
3632, 35preq12d 3893 . . . . 5 ℤ/n mulGrp MndHom mulGrpfld Unit
378, 36csbied 3295 . . . 4 ℤ/n mulGrp MndHom mulGrpfld Unit
385, 37csbied 3295 . . 3 ℤ/n mulGrp MndHom mulGrpfld Unit
39 dchrval.n . . 3
40 prex 4408 . . . 4
4140a1i 11 . . 3
423, 38, 39, 41fvmptd 5812 . 2 DChr
431, 42syl5eq 2482 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  crab 2711  cvv 2958  csb 3253   cdif 3319   wss 3322  csn 3816  cpr 3817  cop 3819   cmpt 4268   cxp 4878   cres 4882  cfv 5456  (class class class)co 6083   cof 6305  cc0 8992   cmul 8997  cn 10002  cnx 13468  cbs 13471   cplusg 13531   MndHom cmhm 14738  mulGrpcmgp 15650  Unitcui 15746  ℂfldccnfld 16705  ℤ/nℤczn 16783  DChrcdchr 21018 This theorem is referenced by:  dchrbas  21021  dchrplusg  21033 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-dchr 21019
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