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Mirrors > Home > MPE Home > Th. List > dchrmulid2 | Structured version Visualization version GIF version |
Description: Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.) |
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)) |
dchrmulid2.t | ⊢ · = (+g‘𝐺) |
dchrmulid2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
dchrmulid2 | ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
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 | dchrmulid2.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 | dchrmulid2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
9 | 1, 3 | dchrrcl 24765 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
11 | 1, 2, 3, 5, 6, 7, 10 | dchr1cl 24776 | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) |
12 | 1, 2, 3, 4, 11, 8 | dchrmul 24773 | . 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘𝑓 · 𝑋)) |
13 | oveq1 6556 | . . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
14 | 13 | eqeq1d 2612 | . . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
15 | oveq1 6556 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
16 | 15 | eqeq1d 2612 | . . . . 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 24767 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
18 | 17 | ffvelrnda 6267 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ) |
19 | 18 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ) |
20 | 19 | mulid2d 9937 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
21 | 0cn 9911 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
22 | 21 | mul02i 10104 | . . . . . 6 ⊢ (0 · 0) = 0 |
23 | 1, 2, 5, 6, 10, 3 | dchrelbas2 24762 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)))) |
24 | 8, 23 | mpbid 221 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))) |
25 | 24 | simprd 478 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
26 | 25 | r19.21bi 2916 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
27 | 26 | necon1bd 2800 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
28 | 27 | imp 444 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) = 0) |
29 | 28 | oveq2d 6565 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0)) |
30 | 22, 29, 28 | 3eqtr4a 2670 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
31 | 14, 16, 20, 30 | ifbothda 4073 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
32 | 31 | mpteq2dva 4672 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
33 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝑍) ∈ V | |
34 | 5, 33 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
36 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
37 | 36, 21 | keepel 4105 | . . . . 5 ⊢ if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ |
38 | 37 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ) |
39 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
40 | 17 | feqmptd 6159 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
41 | 35, 38, 18, 39, 40 | offval2 6812 | . . 3 ⊢ (𝜑 → ( 1 ∘𝑓 · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))) |
42 | 32, 41, 40 | 3eqtr4d 2654 | . 2 ⊢ (𝜑 → ( 1 ∘𝑓 · 𝑋) = 𝑋) |
43 | 12, 42 | eqtrd 2644 | 1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ifcif 4036 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 ℕcn 10897 Basecbs 15695 +gcplusg 15768 MndHom cmhm 17156 mulGrpcmgp 18312 Unitcui 18462 ℂfldccnfld 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-of 6795 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-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: dchrabl 24779 dchr1 24782 |
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