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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 2748 | . . . . . 6 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | |
| 2 | eqid 2748 | . . . . . 6 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁)) | |
| 3 | 1, 2 | znfi 20081 | . . . . 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 25150 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) |
| 7 | simprr 813 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
| 8 | 4, 1, 5, 2, 7 | dchrf 25137 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ) |
| 9 | simprl 811 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
| 10 | 8, 9 | ffvelrnd 6511 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → (𝑥‘𝑎) ∈ ℂ) |
| 11 | 3, 6, 10 | fsumcom 14677 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = Σ𝑥 ∈ 𝐷 Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎)) |
| 12 | eqid 2748 | . . . . . . 7 ⊢ (1r‘(ℤ/nℤ‘𝑁)) = (1r‘(ℤ/nℤ‘𝑁)) | |
| 13 | simpl 474 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑁 ∈ ℕ) | |
| 14 | simpr 479 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
| 15 | 4, 5, 1, 12, 2, 13, 14 | sumdchr2 25165 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 = (1r‘(ℤ/nℤ‘𝑁)), (♯‘𝐷), 0)) |
| 16 | velsn 4325 | . . . . . . 7 ⊢ (𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} ↔ 𝑎 = (1r‘(ℤ/nℤ‘𝑁))) | |
| 17 | ifbi 4239 | . . . . . . 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 2785 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0)) |
| 20 | 19 | sumeq2dv 14603 | . . . 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 2748 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 22 | simpr 479 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
| 23 | 4, 1, 5, 21, 22, 2 | dchrsum 25164 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 = (0g‘𝐺), (ϕ‘𝑁), 0)) |
| 24 | velsn 4325 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
| 25 | ifbi 4239 | . . . . . . 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 2785 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 28 | 27 | sumeq2dv 14603 | . . . 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 2790 | . . 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 11462 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
| 31 | 1 | zncrng 20066 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing) |
| 32 | crngring 18729 | . . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring) | |
| 33 | 2, 12 | ringidcl 18739 | . . . . . 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 4473 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(1r‘(ℤ/nℤ‘𝑁))} ⊆ (Base‘(ℤ/nℤ‘𝑁))) |
| 36 | hashcl 13310 | . . . . . 6 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ0) | |
| 37 | nn0cn 11465 | . . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ0 → (♯‘𝐷) ∈ ℂ) | |
| 38 | 6, 36, 37 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) ∈ ℂ) |
| 39 | 38 | ralrimivw 3093 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) ∈ ℂ) |
| 40 | 3 | olcd 407 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((Base‘(ℤ/nℤ‘𝑁)) ⊆ (ℤ≥‘0) ∨ (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)) |
| 41 | sumss2 14627 | . . . 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 1462 | . . 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 25149 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
| 44 | ablgrp 18369 | . . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 45 | 5, 21 | grpidcl 17622 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐷) |
| 46 | 43, 44, 45 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0g‘𝐺) ∈ 𝐷) |
| 47 | 46 | snssd 4473 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(0g‘𝐺)} ⊆ 𝐷) |
| 48 | phicl 15647 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | |
| 49 | 48 | nncnd 11199 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℂ) |
| 50 | 49 | ralrimivw 3093 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) ∈ ℂ) |
| 51 | 6 | olcd 407 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐷 ⊆ (ℤ≥‘0) ∨ 𝐷 ∈ Fin)) |
| 52 | sumss2 14627 | . . . 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 1462 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 54 | 29, 42, 53 | 3eqtr4d 2792 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁)) |
| 55 | eqidd 2749 | . . . 4 ⊢ (𝑎 = (1r‘(ℤ/nℤ‘𝑁)) → (♯‘𝐷) = (♯‘𝐷)) | |
| 56 | 55 | sumsn 14645 | . . 3 ⊢ (((1r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (♯‘𝐷) ∈ ℂ) → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 57 | 34, 38, 56 | syl2anc 696 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 58 | eqidd 2749 | . . . 4 ⊢ (𝑥 = (0g‘𝐺) → (ϕ‘𝑁) = (ϕ‘𝑁)) | |
| 59 | 58 | sumsn 14645 | . . 3 ⊢ (((0g‘𝐺) ∈ 𝐷 ∧ (ϕ‘𝑁) ∈ ℂ) → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 60 | 46, 49, 59 | syl2anc 696 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 61 | 54, 57, 60 | 3eqtr3d 2790 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) = (ϕ‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ∀wral 3038 ⊆ wss 3703 ifcif 4218 {csn 4309 ‘cfv 6037 Fincfn 8109 ℂcc 10097 0cc0 10099 ℕcn 11183 ℕ0cn0 11455 ℤ≥cuz 11850 ♯chash 13282 Σcsu 14586 ϕcphi 15642 Basecbs 16030 0gc0g 16273 Grpcgrp 17594 Abelcabl 18365 1rcur 18672 Ringcrg 18718 CRingccrg 18719 ℤ/nℤczn 20024 DChrcdchr 25127 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-inf2 8699 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 ax-addf 10178 ax-mulf 10179 |
| This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-iin 4663 df-disj 4761 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-isom 6046 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-rpss 7090 df-om 7219 df-1st 7321 df-2nd 7322 df-supp 7452 df-tpos 7509 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7899 df-ec 7901 df-qs 7905 df-map 8013 df-pm 8014 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8429 df-fi 8470 df-sup 8501 df-inf 8502 df-oi 8568 df-card 8926 df-acn 8929 df-cda 9153 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-7 11247 df-8 11248 df-9 11249 df-n0 11456 df-xnn0 11527 df-z 11541 df-dec 11657 df-uz 11851 df-q 11953 df-rp 11997 df-xneg 12110 df-xadd 12111 df-xmul 12112 df-ioo 12343 df-ioc 12344 df-ico 12345 df-icc 12346 df-fz 12491 df-fzo 12631 df-fl 12758 df-mod 12834 df-seq 12967 df-exp 13026 df-fac 13226 df-bc 13255 df-hash 13283 df-word 13456 df-concat 13458 df-s1 13459 df-shft 13977 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-limsup 14372 df-clim 14389 df-rlim 14390 df-sum 14587 df-ef 14968 df-sin 14970 df-cos 14971 df-pi 14973 df-dvds 15154 df-gcd 15390 df-prm 15559 df-phi 15644 df-pc 15715 df-struct 16032 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-mulr 16128 df-starv 16129 df-sca 16130 df-vsca 16131 df-ip 16132 df-tset 16133 df-ple 16134 df-ds 16137 df-unif 16138 df-hom 16139 df-cco 16140 df-rest 16256 df-topn 16257 df-0g 16275 df-gsum 16276 df-topgen 16277 df-pt 16278 df-prds 16281 df-xrs 16335 df-qtop 16340 df-imas 16341 df-qus 16342 df-xps 16343 df-mre 16419 df-mrc 16420 df-acs 16422 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-mhm 17507 df-submnd 17508 df-grp 17597 df-minusg 17598 df-sbg 17599 df-mulg 17713 df-subg 17763 df-nsg 17764 df-eqg 17765 df-ghm 17830 df-gim 17873 df-ga 17894 df-cntz 17921 df-oppg 17947 df-od 18119 df-gex 18120 df-pgp 18121 df-lsm 18222 df-pj1 18223 df-cmn 18366 df-abl 18367 df-cyg 18451 df-dprd 18565 df-dpj 18566 df-mgp 18661 df-ur 18673 df-ring 18720 df-cring 18721 df-oppr 18794 df-dvdsr 18812 df-unit 18813 df-invr 18843 df-rnghom 18888 df-subrg 18951 df-lmod 19038 df-lss 19106 df-lsp 19145 df-sra 19345 df-rgmod 19346 df-lidl 19347 df-rsp 19348 df-2idl 19405 df-psmet 19911 df-xmet 19912 df-met 19913 df-bl 19914 df-mopn 19915 df-fbas 19916 df-fg 19917 df-cnfld 19920 df-zring 19992 df-zrh 20025 df-zn 20028 df-top 20872 df-topon 20889 df-topsp 20910 df-bases 20923 df-cld 20996 df-ntr 20997 df-cls 20998 df-nei 21075 df-lp 21113 df-perf 21114 df-cn 21204 df-cnp 21205 df-haus 21292 df-tx 21538 df-hmeo 21731 df-fil 21822 df-fm 21914 df-flim 21915 df-flf 21916 df-xms 22297 df-ms 22298 df-tms 22299 df-cncf 22853 df-0p 23607 df-limc 23800 df-dv 23801 df-ply 24114 df-idp 24115 df-coe 24116 df-dgr 24117 df-quot 24216 df-log 24473 df-cxp 24474 df-dchr 25128 |
| This theorem is referenced by: sumdchr 25167 |
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