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Theorem dchreq 27206

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 3909	. . . . 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 2733	. . . . . . . . 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 27198	. . . . . . . 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 2948	. . . . . 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 27198	. . . . . . . 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 2948	. . . . . 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 2771	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘))
23	22	ralrimiva 3126	. 2 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))
24	2, 3, 4, 5, 7	dchrf 27190	. . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
25	24	ffnd 6660	. . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
26	2, 3, 4, 5, 15	dchrf 27190	. . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
27	26	ffnd 6660	. . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
28		eqfnfv 6973	. . . 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 20304	. . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍)
31		undif 4433	. . . . . 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 3292	. . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))
34		ralunb 4148	. . . 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 2113 ≠ wne 2930 ∀wral 3049 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 Fn wfn 6484 ‘cfv 6489 ℂcc 11014 0cc0 11016 Basecbs 17130 Unitcui 20283 ℤ/nℤczn 21449 DChrcdchr 27180

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-addf 11095 ax-mulf 11096

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-ec 8633 df-qs 8637 df-map 8761 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-sup 9336 df-inf 9337 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-nn 12136 df-2 12198 df-3 12199 df-4 12200 df-5 12201 df-6 12202 df-7 12203 df-8 12204 df-9 12205 df-n0 12392 df-z 12479 df-dec 12599 df-uz 12743 df-fz 13418 df-struct 17068 df-sets 17085 df-slot 17103 df-ndx 17115 df-base 17131 df-ress 17152 df-plusg 17184 df-mulr 17185 df-starv 17186 df-sca 17187 df-vsca 17188 df-ip 17189 df-tset 17190 df-ple 17191 df-ds 17193 df-unif 17194 df-0g 17355 df-imas 17422 df-qus 17423 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-mhm 18701 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19046 df-nsg 19047 df-eqg 19048 df-cmn 19704 df-abl 19705 df-mgp 20069 df-rng 20081 df-ur 20110 df-ring 20163 df-cring 20164 df-oppr 20265 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-subrng 20471 df-subrg 20495 df-lmod 20805 df-lss 20875 df-lsp 20915 df-sra 21117 df-rgmod 21118 df-lidl 21155 df-rsp 21156 df-2idl 21197 df-cnfld 21302 df-zring 21394 df-zn 21453 df-dchr 27181

This theorem is referenced by: dchrresb 27207 dchrinv 27209 dchrsum2 27216

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