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Mirrors > Home > MPE Home > Th. List > dchr1re | Structured version Visualization version GIF version |
Description: The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.) |
Ref | Expression |
---|---|
dchr1re.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchr1re.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchr1re.o | ⊢ 1 = (0g‘𝐺) |
dchr1re.b | ⊢ 𝐵 = (Base‘𝑍) |
dchr1re.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
dchr1re | ⊢ (𝜑 → 1 :𝐵⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchr1re.g | . . . 4 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchr1re.z | . . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | eqid 2798 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | dchr1re.b | . . . 4 ⊢ 𝐵 = (Base‘𝑍) | |
5 | dchr1re.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 1 | dchrabl 25838 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
7 | ablgrp 18903 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | dchr1re.o | . . . . . 6 ⊢ 1 = (0g‘𝐺) | |
9 | 3, 8 | grpidcl 18123 | . . . . 5 ⊢ (𝐺 ∈ Grp → 1 ∈ (Base‘𝐺)) |
10 | 5, 6, 7, 9 | 4syl 19 | . . . 4 ⊢ (𝜑 → 1 ∈ (Base‘𝐺)) |
11 | 1, 2, 3, 4, 10 | dchrf 25826 | . . 3 ⊢ (𝜑 → 1 :𝐵⟶ℂ) |
12 | 11 | ffnd 6488 | . 2 ⊢ (𝜑 → 1 Fn 𝐵) |
13 | simpr 488 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) = 0) → ( 1 ‘𝑥) = 0) | |
14 | 0re 10632 | . . . . 5 ⊢ 0 ∈ ℝ | |
15 | 13, 14 | eqeltrdi 2898 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) = 0) → ( 1 ‘𝑥) ∈ ℝ) |
16 | eqid 2798 | . . . . . 6 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
17 | 5 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → 𝑁 ∈ ℕ) |
18 | 10 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (Base‘𝐺)) |
19 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
20 | 1, 2, 3, 4, 16, 18, 19 | dchrn0 25834 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 1 ‘𝑥) ≠ 0 ↔ 𝑥 ∈ (Unit‘𝑍))) |
21 | 20 | biimpa 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)) |
22 | 1, 2, 8, 16, 17, 21 | dchr1 25841 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → ( 1 ‘𝑥) = 1) |
23 | 1re 10630 | . . . . 5 ⊢ 1 ∈ ℝ | |
24 | 22, 23 | eqeltrdi 2898 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → ( 1 ‘𝑥) ∈ ℝ) |
25 | 15, 24 | pm2.61dane 3074 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 ‘𝑥) ∈ ℝ) |
26 | 25 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 ‘𝑥) ∈ ℝ) |
27 | ffnfv 6859 | . 2 ⊢ ( 1 :𝐵⟶ℝ ↔ ( 1 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 ( 1 ‘𝑥) ∈ ℝ)) | |
28 | 12, 26, 27 | sylanbrc 586 | 1 ⊢ (𝜑 → 1 :𝐵⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 ℕcn 11625 Basecbs 16475 0gc0g 16705 Grpcgrp 18095 Abelcabl 18899 Unitcui 19385 ℤ/nℤczn 20196 DChrcdchr 25816 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-imas 16773 df-qus 16774 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-nsg 18269 df-eqg 18270 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-cnfld 20092 df-zring 20164 df-zn 20200 df-dchr 25817 |
This theorem is referenced by: rpvmasumlem 26071 |
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