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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 2733 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | dchr1re.b | . . . 4 ⊢ 𝐵 = (Base‘𝑍) | |
5 | dchr1re.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 1 | dchrabl 26747 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
7 | ablgrp 19648 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | dchr1re.o | . . . . . 6 ⊢ 1 = (0g‘𝐺) | |
9 | 3, 8 | grpidcl 18847 | . . . . 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 26735 | . . 3 ⊢ (𝜑 → 1 :𝐵⟶ℂ) |
12 | 11 | ffnd 6716 | . 2 ⊢ (𝜑 → 1 Fn 𝐵) |
13 | simpr 486 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) = 0) → ( 1 ‘𝑥) = 0) | |
14 | 0re 11213 | . . . . 5 ⊢ 0 ∈ ℝ | |
15 | 13, 14 | eqeltrdi 2842 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) = 0) → ( 1 ‘𝑥) ∈ ℝ) |
16 | eqid 2733 | . . . . . 6 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
17 | 5 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → 𝑁 ∈ ℕ) |
18 | 10 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (Base‘𝐺)) |
19 | simpr 486 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
20 | 1, 2, 3, 4, 16, 18, 19 | dchrn0 26743 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 1 ‘𝑥) ≠ 0 ↔ 𝑥 ∈ (Unit‘𝑍))) |
21 | 20 | biimpa 478 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)) |
22 | 1, 2, 8, 16, 17, 21 | dchr1 26750 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → ( 1 ‘𝑥) = 1) |
23 | 1re 11211 | . . . . 5 ⊢ 1 ∈ ℝ | |
24 | 22, 23 | eqeltrdi 2842 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → ( 1 ‘𝑥) ∈ ℝ) |
25 | 15, 24 | pm2.61dane 3030 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 ‘𝑥) ∈ ℝ) |
26 | 25 | ralrimiva 3147 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 ‘𝑥) ∈ ℝ) |
27 | ffnfv 7115 | . 2 ⊢ ( 1 :𝐵⟶ℝ ↔ ( 1 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 ( 1 ‘𝑥) ∈ ℝ)) | |
28 | 12, 26, 27 | sylanbrc 584 | 1 ⊢ (𝜑 → 1 :𝐵⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 ℕcn 12209 Basecbs 17141 0gc0g 17382 Grpcgrp 18816 Abelcabl 19644 Unitcui 20162 ℤ/nℤczn 21044 DChrcdchr 26725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-ec 8702 df-qs 8706 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-0g 17384 df-imas 17451 df-qus 17452 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-grp 18819 df-minusg 18820 df-sbg 18821 df-subg 18998 df-nsg 18999 df-eqg 19000 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-subrg 20354 df-lmod 20466 df-lss 20536 df-lsp 20576 df-sra 20778 df-rgmod 20779 df-lidl 20780 df-rsp 20781 df-2idl 20850 df-cnfld 20938 df-zring 21011 df-zn 21048 df-dchr 26726 |
This theorem is referenced by: rpvmasumlem 26980 |
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