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Theorem dchrmullid 27224

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 27212	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	1, 2, 3, 5, 6, 7, 10	dchr1cl 27223	. . 3 ⊢ (𝜑 → 1 ∈ 𝐷)
12	1, 2, 3, 4, 11, 8	dchrmul 27220	. 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘_f · 𝑋))
13		oveq1 7368	. . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))
14	13	eqeq1d 2739	. . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)))
15		oveq1 7368	. . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))
16	15	eqeq1d 2739	. . . . 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 27214	. . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
18	17	ffvelcdmda 7031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
20	19	mullidd 11155	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘))
21		0cn 11129	. . . . . . 7 ⊢ 0 ∈ ℂ
22	21	mul02i 11327	. . . . . 6 ⊢ (0 · 0) = 0
23	1, 2, 5, 6, 10, 3	dchrelbas2 27209	. . . . . . . . . . . 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 3229	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
27	26	necon1bd 2951	. . . . . . . 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 7377	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0))
30	22, 29, 28	3eqtr4a 2798	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘))
31	14, 16, 20, 30	ifbothda 4519	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))
32	31	mpteq2dva 5192	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
33	5	fvexi 6849	. . . . 5 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
35		ax-1cn 11089	. . . . . 6 ⊢ 1 ∈ ℂ
36	35, 21	ifcli 4528	. . . . 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 6903	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
40	34, 37, 18, 38, 39	offval2 7645	. . 3 ⊢ (𝜑 → ( 1 ∘_f · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))))
41	32, 40, 39	3eqtr4d 2782	. 2 ⊢ (𝜑 → ( 1 ∘_f · 𝑋) = 𝑋)
42	12, 41	eqtrd 2772	1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3441 ifcif 4480 ↦ cmpt 5180 ‘cfv 6493 (class class class)co 7361 ∘_f cof 7623 ℂcc 11029 0cc0 11031 1c1 11032 · cmul 11036 ℕcn 12150 Basecbs 17141 +_gcplusg 17182 MndHom cmhm 18711 mulGrpcmgp 20080 Unitcui 20296 ℂ_fldccnfld 21314 ℤ/nℤczn 21462 DChrcdchr 27204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108 ax-addf 11110 ax-mulf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-sup 9350 df-inf 9351 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-z 12494 df-dec 12613 df-uz 12757 df-fz 13429 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-starv 17197 df-sca 17198 df-vsca 17199 df-ip 17200 df-tset 17201 df-ple 17202 df-ds 17204 df-unif 17205 df-0g 17366 df-imas 17434 df-qus 17435 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18713 df-grp 18871 df-minusg 18872 df-sbg 18873 df-subg 19058 df-nsg 19059 df-eqg 19060 df-cmn 19716 df-abl 19717 df-mgp 20081 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20278 df-dvdsr 20298 df-unit 20299 df-subrng 20484 df-subrg 20508 df-lmod 20818 df-lss 20888 df-lsp 20928 df-sra 21130 df-rgmod 21131 df-lidl 21168 df-rsp 21169 df-2idl 21210 df-cnfld 21315 df-zring 21407 df-zn 21466 df-dchr 27205

This theorem is referenced by: dchrabl 27226 dchr1 27229

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