dchrn0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dchrn0		Structured version Visualization version GIF version

Theorem dchrn0 24775

Description: A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrmhm.g	⊢ 𝐺 = (DChr‘𝑁)
dchrmhm.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrmhm.b	⊢ 𝐷 = (Base‘𝐺)
dchrn0.b	⊢ 𝐵 = (Base‘𝑍)
dchrn0.u	⊢ 𝑈 = (Unit‘𝑍)
dchrn0.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrn0.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

**Assertion**
Ref	Expression
dchrn0	⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈))

Proof of Theorem dchrn0 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrn0.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		dchrn0.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
3		dchrmhm.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
4		dchrmhm.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
5		dchrn0.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑍)
6		dchrn0.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑍)
7		dchrmhm.b	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
8	3, 7	dchrrcl 24765	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
9	2, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
10	3, 4, 5, 6, 9, 7	dchrelbas2 24762	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))))
11	2, 10	mpbid 221	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)))
12	11	simprd 478	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))
13		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑋‘𝑥) = (𝑋‘𝐴))
14	13	neeq1d 2841	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑋‘𝑥) ≠ 0 ↔ (𝑋‘𝐴) ≠ 0))
15		eleq1 2676	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
16	14, 15	imbi12d 333	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈) ↔ ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈)))
17	16	rspcv 3278	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈) → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈)))
18	1, 12, 17	sylc 63	. . 3 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈))
19	18	imp 444	. 2 ⊢ ((𝜑 ∧ (𝑋‘𝐴) ≠ 0) → 𝐴 ∈ 𝑈)
20		ax-1ne0 9884	. . . . 5 ⊢ 1 ≠ 0
21	20	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 1 ≠ 0)
22	9	nnnn0d 11228	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
23	4	zncrng 19712	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
24		crngring 18381	. . . . . . . 8 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
25	22, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ Ring)
26		eqid 2610	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
27		eqid 2610	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
28		eqid 2610	. . . . . . . 8 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
29	6, 26, 27, 28	unitrinv 18501	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
30	25, 29	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
31	30	fveq2d 6107	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = (𝑋‘(1_r‘𝑍)))
32	11	simpld 474	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
33	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
34	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝐵)
35	6, 26, 5	ringinvcl 18499	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
36	25, 35	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
37		eqid 2610	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
38	37, 5	mgpbas 18318	. . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
39	37, 27	mgpplusg 18316	. . . . . . 7 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
40		eqid 2610	. . . . . . . 8 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
41		cnfldmul 19573	. . . . . . . 8 ⊢ · = (._r‘ℂ_fld)
42	40, 41	mgpplusg 18316	. . . . . . 7 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
43	38, 39, 42	mhmlin 17165	. . . . . 6 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝐴 ∈ 𝐵 ∧ ((inv_r‘𝑍)‘𝐴) ∈ 𝐵) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))))
44	33, 34, 36, 43	syl3anc 1318	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))))
45	37, 28	ringidval 18326	. . . . . . 7 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
46		cnfld1 19590	. . . . . . . 8 ⊢ 1 = (1_r‘ℂ_fld)
47	40, 46	ringidval 18326	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
48	45, 47	mhm0 17166	. . . . . 6 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑋‘(1_r‘𝑍)) = 1)
49	33, 48	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(1_r‘𝑍)) = 1)
50	31, 44, 49	3eqtr3d 2652	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 1)
51		cnfldbas 19571	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
52	40, 51	mgpbas 18318	. . . . . . . 8 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
53	38, 52	mhmf 17163	. . . . . . 7 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → 𝑋:𝐵⟶ℂ)
54	33, 53	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋:𝐵⟶ℂ)
55	54, 36	ffvelrnd 6268	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘((inv_r‘𝑍)‘𝐴)) ∈ ℂ)
56	55	mul02d 10113	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 0)
57	21, 50, 56	3netr4d 2859	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
58		oveq1 6556	. . . 4 ⊢ ((𝑋‘𝐴) = 0 → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
59	58	necon3i 2814	. . 3 ⊢ (((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))) → (𝑋‘𝐴) ≠ 0)
60	57, 59	syl 17	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘𝐴) ≠ 0)
61	19, 60	impbida 873	1 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 ℕcn 10897 ℕ₀cn0 11169 Basecbs 15695 ._rcmulr 15769 MndHom cmhm 17156 mulGrpcmgp 18312 1_rcur 18324 Ringcrg 18370 CRingccrg 18371 Unitcui 18462 inv_rcinvr 18494 ℂ_fldccnfld 19567 ℤ/nℤczn 19670 DChrcdchr 24757

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-nsg 17415 df-eqg 17416 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zn 19674 df-dchr 24758

This theorem is referenced by: dchrinvcl 24778 dchrfi 24780 dchrghm 24781 dchreq 24783 dchrabs 24785 dchrabs2 24787 dchr1re 24788 dchrpt 24792 dchrsum 24794 sum2dchr 24799 rpvmasumlem 24976 dchrisum0flblem1 24997

W3C validator