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| Mirrors > Home > MPE Home > Th. List > dchrhash | Structured version Visualization version GIF version | ||
| Description: There are exactly ϕ(𝑁) Dirichlet characters modulo 𝑁. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
| Ref | Expression |
|---|---|
| sumdchr.g | ⊢ 𝐺 = (DChr‘𝑁) |
| sumdchr.d | ⊢ 𝐷 = (Base‘𝐺) |
| Ref | Expression |
|---|---|
| dchrhash | ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) = (ϕ‘𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2729 | . . . . . 6 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | |
| 2 | eqid 2729 | . . . . . 6 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁)) | |
| 3 | 1, 2 | znfi 21469 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin) |
| 4 | sumdchr.g | . . . . . 6 ⊢ 𝐺 = (DChr‘𝑁) | |
| 5 | sumdchr.d | . . . . . 6 ⊢ 𝐷 = (Base‘𝐺) | |
| 6 | 4, 5 | dchrfi 27166 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) |
| 7 | simprr 772 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
| 8 | 4, 1, 5, 2, 7 | dchrf 27153 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ) |
| 9 | simprl 770 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
| 10 | 8, 9 | ffvelcdmd 7057 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → (𝑥‘𝑎) ∈ ℂ) |
| 11 | 3, 6, 10 | fsumcom 15741 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = Σ𝑥 ∈ 𝐷 Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎)) |
| 12 | eqid 2729 | . . . . . . 7 ⊢ (1r‘(ℤ/nℤ‘𝑁)) = (1r‘(ℤ/nℤ‘𝑁)) | |
| 13 | simpl 482 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑁 ∈ ℕ) | |
| 14 | simpr 484 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
| 15 | 4, 5, 1, 12, 2, 13, 14 | sumdchr2 27181 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 = (1r‘(ℤ/nℤ‘𝑁)), (♯‘𝐷), 0)) |
| 16 | velsn 4605 | . . . . . . 7 ⊢ (𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} ↔ 𝑎 = (1r‘(ℤ/nℤ‘𝑁))) | |
| 17 | ifbi 4511 | . . . . . . 7 ⊢ ((𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} ↔ 𝑎 = (1r‘(ℤ/nℤ‘𝑁))) → if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0) = if(𝑎 = (1r‘(ℤ/nℤ‘𝑁)), (♯‘𝐷), 0)) | |
| 18 | 16, 17 | mp1i 13 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0) = if(𝑎 = (1r‘(ℤ/nℤ‘𝑁)), (♯‘𝐷), 0)) |
| 19 | 15, 18 | eqtr4d 2767 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0)) |
| 20 | 19 | sumeq2dv 15668 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0)) |
| 21 | eqid 2729 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 22 | simpr 484 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
| 23 | 4, 1, 5, 21, 22, 2 | dchrsum 27180 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 = (0g‘𝐺), (ϕ‘𝑁), 0)) |
| 24 | velsn 4605 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
| 25 | ifbi 4511 | . . . . . . 7 ⊢ ((𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) → if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0) = if(𝑥 = (0g‘𝐺), (ϕ‘𝑁), 0)) | |
| 26 | 24, 25 | mp1i 13 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0) = if(𝑥 = (0g‘𝐺), (ϕ‘𝑁), 0)) |
| 27 | 23, 26 | eqtr4d 2767 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 28 | 27 | sumeq2dv 15668 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝐷 Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 29 | 11, 20, 28 | 3eqtr3d 2772 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 30 | nnnn0 12449 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
| 31 | 1 | zncrng 21454 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing) |
| 32 | crngring 20154 | . . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring) | |
| 33 | 2, 12 | ringidcl 20174 | . . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (1r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁))) |
| 34 | 30, 31, 32, 33 | 4syl 19 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (1r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁))) |
| 35 | 34 | snssd 4773 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(1r‘(ℤ/nℤ‘𝑁))} ⊆ (Base‘(ℤ/nℤ‘𝑁))) |
| 36 | hashcl 14321 | . . . . . 6 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ0) | |
| 37 | nn0cn 12452 | . . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ0 → (♯‘𝐷) ∈ ℂ) | |
| 38 | 6, 36, 37 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) ∈ ℂ) |
| 39 | 38 | ralrimivw 3129 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) ∈ ℂ) |
| 40 | 3 | olcd 874 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((Base‘(ℤ/nℤ‘𝑁)) ⊆ (ℤ≥‘0) ∨ (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)) |
| 41 | sumss2 15692 | . . . 4 ⊢ ((({(1r‘(ℤ/nℤ‘𝑁))} ⊆ (Base‘(ℤ/nℤ‘𝑁)) ∧ ∀𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) ∈ ℂ) ∧ ((Base‘(ℤ/nℤ‘𝑁)) ⊆ (ℤ≥‘0) ∨ (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)) → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0)) | |
| 42 | 35, 39, 40, 41 | syl21anc 837 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0)) |
| 43 | 4 | dchrabl 27165 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
| 44 | ablgrp 19715 | . . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 45 | 5, 21 | grpidcl 18897 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐷) |
| 46 | 43, 44, 45 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0g‘𝐺) ∈ 𝐷) |
| 47 | 46 | snssd 4773 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(0g‘𝐺)} ⊆ 𝐷) |
| 48 | phicl 16739 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | |
| 49 | 48 | nncnd 12202 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℂ) |
| 50 | 49 | ralrimivw 3129 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) ∈ ℂ) |
| 51 | 6 | olcd 874 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐷 ⊆ (ℤ≥‘0) ∨ 𝐷 ∈ Fin)) |
| 52 | sumss2 15692 | . . . 4 ⊢ ((({(0g‘𝐺)} ⊆ 𝐷 ∧ ∀𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) ∈ ℂ) ∧ (𝐷 ⊆ (ℤ≥‘0) ∨ 𝐷 ∈ Fin)) → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) | |
| 53 | 47, 50, 51, 52 | syl21anc 837 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 54 | 29, 42, 53 | 3eqtr4d 2774 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁)) |
| 55 | eqidd 2730 | . . . 4 ⊢ (𝑎 = (1r‘(ℤ/nℤ‘𝑁)) → (♯‘𝐷) = (♯‘𝐷)) | |
| 56 | 55 | sumsn 15712 | . . 3 ⊢ (((1r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (♯‘𝐷) ∈ ℂ) → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 57 | 34, 38, 56 | syl2anc 584 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 58 | eqidd 2730 | . . . 4 ⊢ (𝑥 = (0g‘𝐺) → (ϕ‘𝑁) = (ϕ‘𝑁)) | |
| 59 | 58 | sumsn 15712 | . . 3 ⊢ (((0g‘𝐺) ∈ 𝐷 ∧ (ϕ‘𝑁) ∈ ℂ) → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 60 | 46, 49, 59 | syl2anc 584 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 61 | 54, 57, 60 | 3eqtr3d 2772 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) = (ϕ‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3914 ifcif 4488 {csn 4589 ‘cfv 6511 Fincfn 8918 ℂcc 11066 0cc0 11068 ℕcn 12186 ℕ0cn0 12442 ℤ≥cuz 12793 ♯chash 14295 Σcsu 15652 ϕcphi 16734 Basecbs 17179 0gc0g 17402 Grpcgrp 18865 Abelcabl 19711 1rcur 20090 Ringcrg 20142 CRingccrg 20143 ℤ/nℤczn 21412 DChrcdchr 27143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-rpss 7699 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-word 14479 df-concat 14536 df-s1 14561 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-dvds 16223 df-gcd 16465 df-prm 16642 df-phi 16736 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-qus 17472 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145 df-gim 19191 df-ga 19222 df-cntz 19249 df-oppg 19278 df-od 19458 df-gex 19459 df-pgp 19460 df-lsm 19566 df-pj1 19567 df-cmn 19712 df-abl 19713 df-cyg 19808 df-dprd 19927 df-dpj 19928 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-lmod 20768 df-lss 20838 df-lsp 20878 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-2idl 21160 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-zn 21416 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-0p 25571 df-limc 25767 df-dv 25768 df-ply 26093 df-idp 26094 df-coe 26095 df-dgr 26096 df-quot 26199 df-log 26465 df-cxp 26466 df-dchr 27144 |
| This theorem is referenced by: sumdchr 27183 |
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