dchrzrh1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dchrzrh1		Structured version Visualization version GIF version

Theorem dchrzrh1 27205

Description: Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.)

**Hypotheses**
Ref	Expression
dchrmhm.g	⊢ 𝐺 = (DChr‘𝑁)
dchrmhm.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrmhm.b	⊢ 𝐷 = (Base‘𝐺)
dchrelbas4.l	⊢ 𝐿 = (ℤRHom‘𝑍)
dchrzrh1.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)

**Assertion**
Ref	Expression
dchrzrh1	⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1)

**Proof of Theorem dchrzrh1**
Step	Hyp	Ref	Expression
1		dchrzrh1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
2		dchrmhm.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
3		dchrmhm.b	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
4	2, 3	dchrrcl 27201	. . . . . 6 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnnn0d 12572	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		dchrmhm.z	. . . . 5 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
8	7	zncrng 21492	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9		crngring 20199	. . . 4 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
10		dchrelbas4.l	. . . . 5 ⊢ 𝐿 = (ℤRHom‘𝑍)
11		eqid 2728	. . . . 5 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
12	10, 11	zrh1 21452	. . . 4 ⊢ (𝑍 ∈ Ring → (𝐿‘1) = (1_r‘𝑍))
13	6, 8, 9, 12	4syl 19	. . 3 ⊢ (𝜑 → (𝐿‘1) = (1_r‘𝑍))
14	13	fveq2d 6906	. 2 ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = (𝑋‘(1_r‘𝑍)))
15	2, 7, 3	dchrmhm 27202	. . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
16	15, 1	sselid 3980	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
17		eqid 2728	. . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
18	17, 11	ringidval 20137	. . . 4 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
19		eqid 2728	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
20		cnfld1 21335	. . . . 5 ⊢ 1 = (1_r‘ℂ_fld)
21	19, 20	ringidval 20137	. . . 4 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
22	18, 21	mhm0 18760	. . 3 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑋‘(1_r‘𝑍)) = 1)
23	16, 22	syl 17	. 2 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
24	14, 23	eqtrd 2768	1 ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 1c1 11149 ℕcn 12252 ℕ₀cn0 12512 Basecbs 17189 MndHom cmhm 18747 mulGrpcmgp 20088 1_rcur 20135 Ringcrg 20187 CRingccrg 20188 ℂ_fldccnfld 21293 ℤRHomczrh 21439 ℤ/nℤczn 21442 DChrcdchr 27193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-addf 11227 ax-mulf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-ec 8735 df-qs 8739 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-seq 14009 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-0g 17432 df-imas 17499 df-qus 17500 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-nsg 19093 df-eqg 19094 df-ghm 19182 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-oppr 20287 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-lmod 20759 df-lss 20830 df-lsp 20870 df-sra 21072 df-rgmod 21073 df-lidl 21118 df-rsp 21119 df-2idl 21158 df-cnfld 21294 df-zring 21387 df-zrh 21443 df-zn 21446 df-dchr 27194

This theorem is referenced by: dchrmusum2 27455 dchrvmasum2lem 27457

W3C validator