dchreq - Metamath Proof Explorer

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Theorem dchreq 27534

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 2760	. . . . . . . . 9 ⊢ (Base‘𝑍) = (Base‘𝑍)
6		dchrresb.u	. . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑍)
7		dchrresb.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
8	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑋 ∈ 𝐷)
9		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑘 ∈ (Base‘𝑍))
10	2, 3, 4, 5, 6, 8, 9	dchrn0 27526	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
11	10	biimpd 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
12	11	necon1bd 2973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0))
13	12	impr 460	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑋‘𝑘) = 0)
14	1, 13	sylan2b 606	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = 0)
15		dchrresb.Y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑌 ∈ 𝐷)
17	2, 3, 4, 5, 6, 16, 9	dchrn0 27526	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
18	17	biimpd 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
19	18	necon1bd 2973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑌‘𝑘) = 0))
20	19	impr 460	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑌‘𝑘) = 0)
21	1, 20	sylan2b 606	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑌‘𝑘) = 0)
22	14, 21	eqtr4d 2798	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘))
23	22	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))
24	2, 3, 4, 5, 7	dchrf 27518	. . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
25	24	ffnd 6699	. . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
26	2, 3, 4, 5, 15	dchrf 27518	. . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ)
27	26	ffnd 6699	. . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍))
28		eqfnfv 7018	. . . 4 ⊢ ((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)))
29	25, 27, 28	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)))
30	5, 6	unitss 20553	. . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍)
31		undif 4438	. . . . . 6 ⊢ (𝑈 ⊆ (Base‘𝑍) ↔ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍))
32	30, 31	mpbi 233	. . . . 5 ⊢ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍)
33	32	raleqi 3317	. . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))
34		ralunb 4143	. . . 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 280	. . 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 290	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))))
37	23, 36	mpbiran2d 721	1 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 Fn wfn 6523 ‘cfv 6528 ℂcc 11155 0cc0 11157 Basecbs 17334 Unitcui 20532 ℤ/nℤczn 21755 DChrcdchr 27508

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234 ax-addf 11236 ax-mulf 11237

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7862 df-1st 7985 df-2nd 7986 df-tpos 8222 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-er 8696 df-ec 8698 df-qs 8702 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-nn 12291 df-2 12360 df-3 12361 df-4 12362 df-5 12363 df-6 12364 df-7 12365 df-8 12366 df-9 12367 df-n0 12562 df-z 12649 df-dec 12770 df-uz 12921 df-fz 13595 df-struct 17272 df-sets 17289 df-slot 17307 df-ndx 17319 df-base 17335 df-ress 17356 df-plusg 17388 df-mulr 17389 df-starv 17390 df-sca 17391 df-vsca 17392 df-ip 17393 df-tset 17394 df-ple 17395 df-ds 17397 df-unif 17398 df-0g 17559 df-imas 17627 df-qus 17628 df-mgm 18763 df-sgrp 18855 df-mnd 18871 df-mhm 18925 df-grp 19094 df-minusg 19095 df-sbg 19096 df-subg 19280 df-nsg 19281 df-eqg 19282 df-cmn 19943 df-abl 19944 df-mgp 20308 df-rng 20322 df-ur 20355 df-ring 20408 df-cring 20409 df-oppr 20514 df-dvdsr 20534 df-unit 20535 df-invr 20565 df-subrng 20745 df-subrg 20769 df-lmod 21084 df-lss 21154 df-lsp 21194 df-sra 21395 df-rgmod 21396 df-lidl 21433 df-rsp 21434 df-2idl 21490 df-cnfld 21626 df-zring 21700 df-zn 21759 df-dchr 27509

This theorem is used by: dchrresb 27535 dchrinv 27537 dchrsum2 27544

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