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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 2823 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | dchr1re.b | . . . 4 ⊢ 𝐵 = (Base‘𝑍) | |
5 | dchr1re.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 1 | dchrabl 25832 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
7 | ablgrp 18913 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | dchr1re.o | . . . . . 6 ⊢ 1 = (0g‘𝐺) | |
9 | 3, 8 | grpidcl 18133 | . . . . 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 25820 | . . 3 ⊢ (𝜑 → 1 :𝐵⟶ℂ) |
12 | 11 | ffnd 6517 | . 2 ⊢ (𝜑 → 1 Fn 𝐵) |
13 | simpr 487 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) = 0) → ( 1 ‘𝑥) = 0) | |
14 | 0re 10645 | . . . . 5 ⊢ 0 ∈ ℝ | |
15 | 13, 14 | eqeltrdi 2923 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) = 0) → ( 1 ‘𝑥) ∈ ℝ) |
16 | eqid 2823 | . . . . . 6 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
17 | 5 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → 𝑁 ∈ ℕ) |
18 | 10 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (Base‘𝐺)) |
19 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
20 | 1, 2, 3, 4, 16, 18, 19 | dchrn0 25828 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 1 ‘𝑥) ≠ 0 ↔ 𝑥 ∈ (Unit‘𝑍))) |
21 | 20 | biimpa 479 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → 𝑥 ∈ (Unit‘𝑍)) |
22 | 1, 2, 8, 16, 17, 21 | dchr1 25835 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → ( 1 ‘𝑥) = 1) |
23 | 1re 10643 | . . . . 5 ⊢ 1 ∈ ℝ | |
24 | 22, 23 | eqeltrdi 2923 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ( 1 ‘𝑥) ≠ 0) → ( 1 ‘𝑥) ∈ ℝ) |
25 | 15, 24 | pm2.61dane 3106 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 ‘𝑥) ∈ ℝ) |
26 | 25 | ralrimiva 3184 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 ‘𝑥) ∈ ℝ) |
27 | ffnfv 6884 | . 2 ⊢ ( 1 :𝐵⟶ℝ ↔ ( 1 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 ( 1 ‘𝑥) ∈ ℝ)) | |
28 | 12, 26, 27 | sylanbrc 585 | 1 ⊢ (𝜑 → 1 :𝐵⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 ℕcn 11640 Basecbs 16485 0gc0g 16715 Grpcgrp 18105 Abelcabl 18909 Unitcui 19391 ℤ/nℤczn 20652 DChrcdchr 25810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-nsg 18279 df-eqg 18280 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-cnfld 20548 df-zring 20620 df-zn 20656 df-dchr 25811 |
This theorem is referenced by: rpvmasumlem 26065 |
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