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Theorem dchrelbas2 25182
Description: A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
Hypotheses
Ref Expression
dchrval.g 𝐺 = (DChr‘𝑁)
dchrval.z 𝑍 = (ℤ/nℤ‘𝑁)
dchrval.b 𝐵 = (Base‘𝑍)
dchrval.u 𝑈 = (Unit‘𝑍)
dchrval.n (𝜑𝑁 ∈ ℕ)
dchrbas.b 𝐷 = (Base‘𝐺)
Assertion
Ref Expression
dchrelbas2 (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑁   𝑥,𝑈   𝜑,𝑥   𝑥,𝑋   𝑥,𝑍
Allowed substitution hints:   𝐷(𝑥)   𝐺(𝑥)

Proof of Theorem dchrelbas2
StepHypRef Expression
1 dchrval.g . . 3 𝐺 = (DChr‘𝑁)
2 dchrval.z . . 3 𝑍 = (ℤ/nℤ‘𝑁)
3 dchrval.b . . 3 𝐵 = (Base‘𝑍)
4 dchrval.u . . 3 𝑈 = (Unit‘𝑍)
5 dchrval.n . . 3 (𝜑𝑁 ∈ ℕ)
6 dchrbas.b . . 3 𝐷 = (Base‘𝐺)
71, 2, 3, 4, 5, 6dchrelbas 25181 . 2 (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ((𝐵𝑈) × {0}) ⊆ 𝑋)))
8 eqid 2770 . . . . . . . . . . 11 (mulGrp‘𝑍) = (mulGrp‘𝑍)
98, 3mgpbas 18702 . . . . . . . . . 10 𝐵 = (Base‘(mulGrp‘𝑍))
10 eqid 2770 . . . . . . . . . . 11 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
11 cnfldbas 19964 . . . . . . . . . . 11 ℂ = (Base‘ℂfld)
1210, 11mgpbas 18702 . . . . . . . . . 10 ℂ = (Base‘(mulGrp‘ℂfld))
139, 12mhmf 17547 . . . . . . . . 9 (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → 𝑋:𝐵⟶ℂ)
1413adantl 467 . . . . . . . 8 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → 𝑋:𝐵⟶ℂ)
15 ffun 6188 . . . . . . . 8 (𝑋:𝐵⟶ℂ → Fun 𝑋)
1614, 15syl 17 . . . . . . 7 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → Fun 𝑋)
17 funssres 6073 . . . . . . 7 ((Fun 𝑋 ∧ ((𝐵𝑈) × {0}) ⊆ 𝑋) → (𝑋 ↾ dom ((𝐵𝑈) × {0})) = ((𝐵𝑈) × {0}))
1816, 17sylan 561 . . . . . 6 (((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) ∧ ((𝐵𝑈) × {0}) ⊆ 𝑋) → (𝑋 ↾ dom ((𝐵𝑈) × {0})) = ((𝐵𝑈) × {0}))
19 simpr 471 . . . . . . 7 (((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) ∧ (𝑋 ↾ dom ((𝐵𝑈) × {0})) = ((𝐵𝑈) × {0})) → (𝑋 ↾ dom ((𝐵𝑈) × {0})) = ((𝐵𝑈) × {0}))
20 resss 5563 . . . . . . 7 (𝑋 ↾ dom ((𝐵𝑈) × {0})) ⊆ 𝑋
2119, 20syl6eqssr 3803 . . . . . 6 (((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) ∧ (𝑋 ↾ dom ((𝐵𝑈) × {0})) = ((𝐵𝑈) × {0})) → ((𝐵𝑈) × {0}) ⊆ 𝑋)
2218, 21impbida 794 . . . . 5 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → (((𝐵𝑈) × {0}) ⊆ 𝑋 ↔ (𝑋 ↾ dom ((𝐵𝑈) × {0})) = ((𝐵𝑈) × {0})))
23 0cn 10233 . . . . . . . . 9 0 ∈ ℂ
24 fconst6g 6234 . . . . . . . . 9 (0 ∈ ℂ → ((𝐵𝑈) × {0}):(𝐵𝑈)⟶ℂ)
2523, 24mp1i 13 . . . . . . . 8 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → ((𝐵𝑈) × {0}):(𝐵𝑈)⟶ℂ)
26 fdm 6191 . . . . . . . 8 (((𝐵𝑈) × {0}):(𝐵𝑈)⟶ℂ → dom ((𝐵𝑈) × {0}) = (𝐵𝑈))
2725, 26syl 17 . . . . . . 7 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → dom ((𝐵𝑈) × {0}) = (𝐵𝑈))
2827reseq2d 5534 . . . . . 6 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → (𝑋 ↾ dom ((𝐵𝑈) × {0})) = (𝑋 ↾ (𝐵𝑈)))
2928eqeq1d 2772 . . . . 5 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → ((𝑋 ↾ dom ((𝐵𝑈) × {0})) = ((𝐵𝑈) × {0}) ↔ (𝑋 ↾ (𝐵𝑈)) = ((𝐵𝑈) × {0})))
3022, 29bitrd 268 . . . 4 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → (((𝐵𝑈) × {0}) ⊆ 𝑋 ↔ (𝑋 ↾ (𝐵𝑈)) = ((𝐵𝑈) × {0})))
31 difss 3886 . . . . . . . 8 (𝐵𝑈) ⊆ 𝐵
32 fssres 6210 . . . . . . . 8 ((𝑋:𝐵⟶ℂ ∧ (𝐵𝑈) ⊆ 𝐵) → (𝑋 ↾ (𝐵𝑈)):(𝐵𝑈)⟶ℂ)
3314, 31, 32sylancl 566 . . . . . . 7 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → (𝑋 ↾ (𝐵𝑈)):(𝐵𝑈)⟶ℂ)
34 ffn 6185 . . . . . . 7 ((𝑋 ↾ (𝐵𝑈)):(𝐵𝑈)⟶ℂ → (𝑋 ↾ (𝐵𝑈)) Fn (𝐵𝑈))
3533, 34syl 17 . . . . . 6 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → (𝑋 ↾ (𝐵𝑈)) Fn (𝐵𝑈))
36 ffn 6185 . . . . . . 7 (((𝐵𝑈) × {0}):(𝐵𝑈)⟶ℂ → ((𝐵𝑈) × {0}) Fn (𝐵𝑈))
3725, 36syl 17 . . . . . 6 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → ((𝐵𝑈) × {0}) Fn (𝐵𝑈))
38 eqfnfv 6454 . . . . . 6 (((𝑋 ↾ (𝐵𝑈)) Fn (𝐵𝑈) ∧ ((𝐵𝑈) × {0}) Fn (𝐵𝑈)) → ((𝑋 ↾ (𝐵𝑈)) = ((𝐵𝑈) × {0}) ↔ ∀𝑥 ∈ (𝐵𝑈)((𝑋 ↾ (𝐵𝑈))‘𝑥) = (((𝐵𝑈) × {0})‘𝑥)))
3935, 37, 38syl2anc 565 . . . . 5 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → ((𝑋 ↾ (𝐵𝑈)) = ((𝐵𝑈) × {0}) ↔ ∀𝑥 ∈ (𝐵𝑈)((𝑋 ↾ (𝐵𝑈))‘𝑥) = (((𝐵𝑈) × {0})‘𝑥)))
40 fvres 6348 . . . . . . . 8 (𝑥 ∈ (𝐵𝑈) → ((𝑋 ↾ (𝐵𝑈))‘𝑥) = (𝑋𝑥))
41 c0ex 10235 . . . . . . . . 9 0 ∈ V
4241fvconst2 6612 . . . . . . . 8 (𝑥 ∈ (𝐵𝑈) → (((𝐵𝑈) × {0})‘𝑥) = 0)
4340, 42eqeq12d 2785 . . . . . . 7 (𝑥 ∈ (𝐵𝑈) → (((𝑋 ↾ (𝐵𝑈))‘𝑥) = (((𝐵𝑈) × {0})‘𝑥) ↔ (𝑋𝑥) = 0))
4443ralbiia 3127 . . . . . 6 (∀𝑥 ∈ (𝐵𝑈)((𝑋 ↾ (𝐵𝑈))‘𝑥) = (((𝐵𝑈) × {0})‘𝑥) ↔ ∀𝑥 ∈ (𝐵𝑈)(𝑋𝑥) = 0)
45 eldif 3731 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑈) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝑈))
4645imbi1i 338 . . . . . . . 8 ((𝑥 ∈ (𝐵𝑈) → (𝑋𝑥) = 0) ↔ ((𝑥𝐵 ∧ ¬ 𝑥𝑈) → (𝑋𝑥) = 0))
47 impexp 437 . . . . . . . 8 (((𝑥𝐵 ∧ ¬ 𝑥𝑈) → (𝑋𝑥) = 0) ↔ (𝑥𝐵 → (¬ 𝑥𝑈 → (𝑋𝑥) = 0)))
48 con1b 347 . . . . . . . . . 10 ((¬ 𝑥𝑈 → (𝑋𝑥) = 0) ↔ (¬ (𝑋𝑥) = 0 → 𝑥𝑈))
49 df-ne 2943 . . . . . . . . . . 11 ((𝑋𝑥) ≠ 0 ↔ ¬ (𝑋𝑥) = 0)
5049imbi1i 338 . . . . . . . . . 10 (((𝑋𝑥) ≠ 0 → 𝑥𝑈) ↔ (¬ (𝑋𝑥) = 0 → 𝑥𝑈))
5148, 50bitr4i 267 . . . . . . . . 9 ((¬ 𝑥𝑈 → (𝑋𝑥) = 0) ↔ ((𝑋𝑥) ≠ 0 → 𝑥𝑈))
5251imbi2i 325 . . . . . . . 8 ((𝑥𝐵 → (¬ 𝑥𝑈 → (𝑋𝑥) = 0)) ↔ (𝑥𝐵 → ((𝑋𝑥) ≠ 0 → 𝑥𝑈)))
5346, 47, 523bitri 286 . . . . . . 7 ((𝑥 ∈ (𝐵𝑈) → (𝑋𝑥) = 0) ↔ (𝑥𝐵 → ((𝑋𝑥) ≠ 0 → 𝑥𝑈)))
5453ralbii2 3126 . . . . . 6 (∀𝑥 ∈ (𝐵𝑈)(𝑋𝑥) = 0 ↔ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈))
5544, 54bitri 264 . . . . 5 (∀𝑥 ∈ (𝐵𝑈)((𝑋 ↾ (𝐵𝑈))‘𝑥) = (((𝐵𝑈) × {0})‘𝑥) ↔ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈))
5639, 55syl6bb 276 . . . 4 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → ((𝑋 ↾ (𝐵𝑈)) = ((𝐵𝑈) × {0}) ↔ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈)))
5730, 56bitrd 268 . . 3 ((𝜑𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) → (((𝐵𝑈) × {0}) ⊆ 𝑋 ↔ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈)))
5857pm5.32da 560 . 2 (𝜑 → ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ((𝐵𝑈) × {0}) ⊆ 𝑋) ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈))))
597, 58bitrd 268 1 (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  cdif 3718  wss 3721  {csn 4314   × cxp 5247  dom cdm 5249  cres 5251  Fun wfun 6025   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6792  cc 10135  0cc0 10137  cn 11221  Basecbs 16063   MndHom cmhm 17540  mulGrpcmgp 18696  Unitcui 18846  fldccnfld 19960  ℤ/nczn 20065  DChrcdchr 25177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-plusg 16161  df-mulr 16162  df-starv 16163  df-tset 16167  df-ple 16168  df-ds 16171  df-unif 16172  df-mhm 17542  df-mgp 18697  df-cnfld 19961  df-dchr 25178
This theorem is referenced by:  dchrelbas3  25183  dchrelbas4  25188  dchrmulcl  25194  dchrn0  25195  dchrmulid2  25197
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