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Theorem dchreq 27196

Description: A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrresb.g	⊢ 𝐺 = (DChr‘𝑁)
dchrresb.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrresb.b	⊢ 𝐷 = (Base‘𝐺)
dchrresb.u	⊢ 𝑈 = (Unit‘𝑍)
dchrresb.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrresb.Y	⊢ (𝜑 → 𝑌 ∈ 𝐷)

**Assertion**
Ref	Expression
dchreq	⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘)))

Distinct variable groups: 𝜑,𝑘 𝑈,𝑘 𝑘,𝑋 𝑘,𝑌 𝑘,𝑍 Allowed substitution hints: 𝐷(𝑘) 𝐺(𝑘) 𝑁(𝑘)

**Proof of Theorem dchreq**
Step	Hyp	Ref	Expression
1		eldif 3907	. . . . 5 ⊢ (𝑘 ∈ ((Base‘𝑍) ∖ 𝑈) ↔ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈))
2		dchrresb.g	. . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁)
3		dchrresb.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
4		dchrresb.b	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
5		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝑍) = (Base‘𝑍)
6		dchrresb.u	. . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑍)
7		dchrresb.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑋 ∈ 𝐷)
9		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑘 ∈ (Base‘𝑍))
10	2, 3, 4, 5, 6, 8, 9	dchrn0 27188	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
11	10	biimpd 229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
12	11	necon1bd 2946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0))
13	12	impr 454	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑋‘𝑘) = 0)
14	1, 13	sylan2b 594	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = 0)
15		dchrresb.Y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
16	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑌 ∈ 𝐷)
17	2, 3, 4, 5, 6, 16, 9	dchrn0 27188	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
18	17	biimpd 229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
19	18	necon1bd 2946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑌‘𝑘) = 0))
20	19	impr 454	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑌‘𝑘) = 0)
21	1, 20	sylan2b 594	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑌‘𝑘) = 0)
22	14, 21	eqtr4d 2769	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘))
23	22	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))
24	2, 3, 4, 5, 7	dchrf 27180	. . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
25	24	ffnd 6652	. . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
26	2, 3, 4, 5, 15	dchrf 27180	. . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
27	26	ffnd 6652	. . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
28		eqfnfv 6964	. . . 4 ⊢ ((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)))
29	25, 27, 28	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)))
30	5, 6	unitss 20294	. . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍)
31		undif 4429	. . . . . 6 ⊢ (𝑈 ⊆ (Base‘𝑍) ↔ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍))
32	30, 31	mpbi 230	. . . . 5 ⊢ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍)
33	32	raleqi 3290	. . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))
34		ralunb 4144	. . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)))
35	33, 34	bitr3i 277	. . 3 ⊢ (∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘) ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)))
36	29, 35	bitrdi 287	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))))
37	23, 36	mpbiran2d 708	1 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∖ cdif 3894 ∪ cun 3895 ⊆ wss 3897 Fn wfn 6476 ‘cfv 6481 ℂcc 11004 0cc0 11006 Basecbs 17120 Unitcui 20273 ℤ/nℤczn 21439 DChrcdchr 27170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-0g 17345 df-imas 17412 df-qus 17413 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-nsg 19037 df-eqg 19038 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-subrng 20461 df-subrg 20485 df-lmod 20795 df-lss 20865 df-lsp 20905 df-sra 21107 df-rgmod 21108 df-lidl 21145 df-rsp 21146 df-2idl 21187 df-cnfld 21292 df-zring 21384 df-zn 21443 df-dchr 27171

This theorem is referenced by: dchrresb 27197 dchrinv 27199 dchrsum2 27206

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