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Theorem dchrmullid 27179

Description: Left identity for the principal Dirichlet character. (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)
dchr1cl.o	⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0))
dchrmullid.t	⊢ · = (+_g‘𝐺)
dchrmullid.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)

**Assertion**
Ref	Expression
dchrmullid	⊢ (𝜑 → ( 1 · 𝑋) = 𝑋)

Distinct variable groups: 𝐵,𝑘 𝑈,𝑘 𝑘,𝑁 𝜑,𝑘 𝑘,𝑋 𝑘,𝑍 Allowed substitution hints: 𝐷(𝑘) · (𝑘) 1 (𝑘) 𝐺(𝑘)

**Proof of Theorem dchrmullid**
Step	Hyp	Ref	Expression
1		dchrmhm.g	. . 3 ⊢ 𝐺 = (DChr‘𝑁)
2		dchrmhm.z	. . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
3		dchrmhm.b	. . 3 ⊢ 𝐷 = (Base‘𝐺)
4		dchrmullid.t	. . 3 ⊢ · = (+_g‘𝐺)
5		dchrn0.b	. . . 4 ⊢ 𝐵 = (Base‘𝑍)
6		dchrn0.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑍)
7		dchr1cl.o	. . . 4 ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0))
8		dchrmullid.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
9	1, 3	dchrrcl 27167	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	1, 2, 3, 5, 6, 7, 10	dchr1cl 27178	. . 3 ⊢ (𝜑 → 1 ∈ 𝐷)
12	1, 2, 3, 4, 11, 8	dchrmul 27175	. 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘_f · 𝑋))
13		oveq1 7360	. . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))
14	13	eqeq1d 2731	. . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)))
15		oveq1 7360	. . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))
16	15	eqeq1d 2731	. . . . 5 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → ((0 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)))
17	1, 2, 3, 5, 8	dchrf 27169	. . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
18	17	ffvelcdmda 7022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
20	19	mullidd 11152	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘))
21		0cn 11126	. . . . . . 7 ⊢ 0 ∈ ℂ
22	21	mul02i 11323	. . . . . 6 ⊢ (0 · 0) = 0
23	1, 2, 5, 6, 10, 3	dchrelbas2 27164	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))))
24	8, 23	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)))
25	24	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
26	25	r19.21bi 3221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
27	26	necon1bd 2943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0))
28	27	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) = 0)
29	28	oveq2d 7369	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0))
30	22, 29, 28	3eqtr4a 2790	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘))
31	14, 16, 20, 30	ifbothda 4517	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))
32	31	mpteq2dva 5188	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
33	5	fvexi 6840	. . . . 5 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
35		ax-1cn 11086	. . . . . 6 ⊢ 1 ∈ ℂ
36	35, 21	ifcli 4526	. . . . 5 ⊢ if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ
37	36	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ)
38	7	a1i 11	. . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)))
39	17	feqmptd 6895	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
40	34, 37, 18, 38, 39	offval2 7637	. . 3 ⊢ (𝜑 → ( 1 ∘_f · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))))
41	32, 40, 39	3eqtr4d 2774	. 2 ⊢ (𝜑 → ( 1 ∘_f · 𝑋) = 𝑋)
42	12, 41	eqtrd 2764	1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3438 ifcif 4478 ↦ cmpt 5176 ‘cfv 6486 (class class class)co 7353 ∘_f cof 7615 ℂcc 11026 0cc0 11028 1c1 11029 · cmul 11033 ℕcn 12146 Basecbs 17138 +_gcplusg 17179 MndHom cmhm 18673 mulGrpcmgp 20043 Unitcui 20258 ℂ_fldccnfld 21279 ℤ/nℤczn 21427 DChrcdchr 27159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-ec 8634 df-qs 8638 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17363 df-imas 17430 df-qus 17431 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-nsg 19021 df-eqg 19022 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-subrng 20449 df-subrg 20473 df-lmod 20783 df-lss 20853 df-lsp 20893 df-sra 21095 df-rgmod 21096 df-lidl 21133 df-rsp 21134 df-2idl 21175 df-cnfld 21280 df-zring 21372 df-zn 21431 df-dchr 27160

This theorem is referenced by: dchrabl 27181 dchr1 27184

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