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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 2737 | . . . . . 6 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | |
| 2 | eqid 2737 | . . . . . 6 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁)) | |
| 3 | 1, 2 | znfi 21519 | . . . . 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 27227 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) |
| 7 | simprr 773 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
| 8 | 4, 1, 5, 2, 7 | dchrf 27214 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ) |
| 9 | simprl 771 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
| 10 | 8, 9 | ffvelcdmd 7032 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → (𝑥‘𝑎) ∈ ℂ) |
| 11 | 3, 6, 10 | fsumcom 15703 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = Σ𝑥 ∈ 𝐷 Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎)) |
| 12 | eqid 2737 | . . . . . . 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 27242 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 = (1r‘(ℤ/nℤ‘𝑁)), (♯‘𝐷), 0)) |
| 16 | velsn 4597 | . . . . . . 7 ⊢ (𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} ↔ 𝑎 = (1r‘(ℤ/nℤ‘𝑁))) | |
| 17 | ifbi 4503 | . . . . . . 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 2775 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0)) |
| 20 | 19 | sumeq2dv 15630 | . . . 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 2737 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 22 | simpr 484 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
| 23 | 4, 1, 5, 21, 22, 2 | dchrsum 27241 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 = (0g‘𝐺), (ϕ‘𝑁), 0)) |
| 24 | velsn 4597 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
| 25 | ifbi 4503 | . . . . . . 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 2775 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 28 | 27 | sumeq2dv 15630 | . . . 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 2780 | . . 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 12413 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
| 31 | 1 | zncrng 21504 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing) |
| 32 | crngring 20185 | . . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring) | |
| 33 | 2, 12 | ringidcl 20205 | . . . . . 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 4766 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(1r‘(ℤ/nℤ‘𝑁))} ⊆ (Base‘(ℤ/nℤ‘𝑁))) |
| 36 | hashcl 14284 | . . . . . 6 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ0) | |
| 37 | nn0cn 12416 | . . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ0 → (♯‘𝐷) ∈ ℂ) | |
| 38 | 6, 36, 37 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) ∈ ℂ) |
| 39 | 38 | ralrimivw 3133 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) ∈ ℂ) |
| 40 | 3 | olcd 875 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((Base‘(ℤ/nℤ‘𝑁)) ⊆ (ℤ≥‘0) ∨ (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)) |
| 41 | sumss2 15654 | . . . 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 838 | . . 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 27226 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
| 44 | ablgrp 19719 | . . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 45 | 5, 21 | grpidcl 18900 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐷) |
| 46 | 43, 44, 45 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0g‘𝐺) ∈ 𝐷) |
| 47 | 46 | snssd 4766 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(0g‘𝐺)} ⊆ 𝐷) |
| 48 | phicl 16701 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | |
| 49 | 48 | nncnd 12166 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℂ) |
| 50 | 49 | ralrimivw 3133 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) ∈ ℂ) |
| 51 | 6 | olcd 875 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐷 ⊆ (ℤ≥‘0) ∨ 𝐷 ∈ Fin)) |
| 52 | sumss2 15654 | . . . 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 838 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 54 | 29, 42, 53 | 3eqtr4d 2782 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁)) |
| 55 | eqidd 2738 | . . . 4 ⊢ (𝑎 = (1r‘(ℤ/nℤ‘𝑁)) → (♯‘𝐷) = (♯‘𝐷)) | |
| 56 | 55 | sumsn 15674 | . . 3 ⊢ (((1r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (♯‘𝐷) ∈ ℂ) → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 57 | 34, 38, 56 | syl2anc 585 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 58 | eqidd 2738 | . . . 4 ⊢ (𝑥 = (0g‘𝐺) → (ϕ‘𝑁) = (ϕ‘𝑁)) | |
| 59 | 58 | sumsn 15674 | . . 3 ⊢ (((0g‘𝐺) ∈ 𝐷 ∧ (ϕ‘𝑁) ∈ ℂ) → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 60 | 46, 49, 59 | syl2anc 585 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 61 | 54, 57, 60 | 3eqtr3d 2780 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) = (ϕ‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3902 ifcif 4480 {csn 4581 ‘cfv 6493 Fincfn 8888 ℂcc 11029 0cc0 11031 ℕcn 12150 ℕ0cn0 12406 ℤ≥cuz 12756 ♯chash 14258 Σcsu 15614 ϕcphi 16696 Basecbs 17141 0gc0g 17364 Grpcgrp 18868 Abelcabl 19715 1rcur 20121 Ringcrg 20173 CRingccrg 20174 ℤ/nℤczn 21462 DChrcdchr 27204 |
| 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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-inf2 9555 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108 ax-pre-sup 11109 ax-addf 11110 ax-mulf 11111 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-rpss 7671 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-omul 8405 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 df-dju 9818 df-card 9856 df-acn 9859 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-xnn0 12480 df-z 12494 df-dec 12613 df-uz 12757 df-q 12867 df-rp 12911 df-xneg 13031 df-xadd 13032 df-xmul 13033 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13717 df-mod 13795 df-seq 13930 df-exp 13990 df-fac 14202 df-bc 14231 df-hash 14259 df-word 14442 df-concat 14499 df-s1 14525 df-shft 14995 df-cj 15027 df-re 15028 df-im 15029 df-sqrt 15163 df-abs 15164 df-limsup 15399 df-clim 15416 df-rlim 15417 df-sum 15615 df-ef 15995 df-sin 15997 df-cos 15998 df-pi 16000 df-dvds 16185 df-gcd 16427 df-prm 16604 df-phi 16698 df-pc 16770 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-starv 17197 df-sca 17198 df-vsca 17199 df-ip 17200 df-tset 17201 df-ple 17202 df-ds 17204 df-unif 17205 df-hom 17206 df-cco 17207 df-rest 17347 df-topn 17348 df-0g 17366 df-gsum 17367 df-topgen 17368 df-pt 17369 df-prds 17372 df-xrs 17428 df-qtop 17433 df-imas 17434 df-qus 17435 df-xps 17436 df-mre 17510 df-mrc 17511 df-acs 17513 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18713 df-submnd 18714 df-grp 18871 df-minusg 18872 df-sbg 18873 df-mulg 19003 df-subg 19058 df-nsg 19059 df-eqg 19060 df-ghm 19147 df-gim 19193 df-ga 19224 df-cntz 19251 df-oppg 19280 df-od 19462 df-gex 19463 df-pgp 19464 df-lsm 19570 df-pj1 19571 df-cmn 19716 df-abl 19717 df-cyg 19812 df-dprd 19931 df-dpj 19932 df-mgp 20081 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20278 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-rhm 20413 df-subrng 20484 df-subrg 20508 df-lmod 20818 df-lss 20888 df-lsp 20928 df-sra 21130 df-rgmod 21131 df-lidl 21168 df-rsp 21169 df-2idl 21210 df-psmet 21306 df-xmet 21307 df-met 21308 df-bl 21309 df-mopn 21310 df-fbas 21311 df-fg 21312 df-cnfld 21315 df-zring 21407 df-zrh 21463 df-zn 21466 df-top 22843 df-topon 22860 df-topsp 22882 df-bases 22895 df-cld 22968 df-ntr 22969 df-cls 22970 df-nei 23047 df-lp 23085 df-perf 23086 df-cn 23176 df-cnp 23177 df-haus 23264 df-tx 23511 df-hmeo 23704 df-fil 23795 df-fm 23887 df-flim 23888 df-flf 23889 df-xms 24269 df-ms 24270 df-tms 24271 df-cncf 24832 df-0p 25632 df-limc 25828 df-dv 25829 df-ply 26154 df-idp 26155 df-coe 26156 df-dgr 26157 df-quot 26260 df-log 26526 df-cxp 26527 df-dchr 27205 |
| This theorem is referenced by: sumdchr 27244 |
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