dchrmullid - Metamath Proof Explorer

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

Theorem dchrmullid 27392

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 27380	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	1, 2, 3, 5, 6, 7, 10	dchr1cl 27391	. . 3 ⊢ (𝜑 → 1 ∈ 𝐷)
12	1, 2, 3, 4, 11, 8	dchrmul 27388	. 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘_f · 𝑋))
13		oveq1 7417	. . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))
14	13	eqeq1d 2763	. . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)))
15		oveq1 7417	. . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))
16	15	eqeq1d 2763	. . . . 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 27382	. . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
18	17	ffvelcdmda 7079	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
19	18	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
20	19	mullidd 11226	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘))
21		0cn 11197	. . . . . . 7 ⊢ 0 ∈ ℂ
22	21	mul02i 11398	. . . . . 6 ⊢ (0 · 0) = 0
23	1, 2, 5, 6, 10, 3	dchrelbas2 27377	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))))
24	8, 23	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)))
25	24	simprd 500	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
26	25	r19.21bi 3255	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))
27	26	necon1bd 2974	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0))
28	27	imp 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) = 0)
29	28	oveq2d 7426	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0))
30	22, 29, 28	3eqtr4a 2822	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘))
31	14, 16, 20, 30	ifbothda 4525	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))
32	31	mpteq2dva 5203	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
33	5	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
35		ax-1cn 11157	. . . . . 6 ⊢ 1 ∈ ℂ
36	35, 21	ifcli 4534	. . . . 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 6949	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
40	34, 37, 18, 38, 39	offval2 7694	. . 3 ⊢ (𝜑 → ( 1 ∘_f · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))))
41	32, 40, 39	3eqtr4d 2806	. 2 ⊢ (𝜑 → ( 1 ∘_f · 𝑋) = 𝑋)
42	12, 41	eqtrd 2796	1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ∀wral 3077 Vcvv 3453 ifcif 4486 ↦ cmpt 5191 ‘cfv 6536 (class class class)co 7410 ∘_f cof 7672 ℂcc 11097 0cc0 11099 1c1 11100 · cmul 11104 ℕcn 12232 Basecbs 17268 +_gcplusg 17309 MndHom cmhm 18838 mulGrpcmgp 20215 Unitcui 20436 ℂ_fldccnfld 21501 ℤ/nℤczn 21631 DChrcdchr 27372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-addf 11178 ax-mulf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-ec 8695 df-qs 8699 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-inf 9402 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-fz 13535 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-0g 17493 df-imas 17561 df-qus 17562 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-grp 19002 df-minusg 19003 df-sbg 19004 df-subg 19188 df-nsg 19189 df-eqg 19190 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-subrng 20630 df-subrg 20654 df-lmod 20962 df-lss 21032 df-lsp 21072 df-sra 21273 df-rgmod 21274 df-lidl 21311 df-rsp 21312 df-2idl 21368 df-cnfld 21502 df-zring 21576 df-zn 21635 df-dchr 27373

This theorem is referenced by: dchrabl 27394 dchr1 27397

W3C validator