Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 2771 | . . . . . 6 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | |
| 2 | eqid 2771 | . . . . . 6 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁)) | |
| 3 | 1, 2 | znfi 20423 | . . . . 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 25548 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) |
| 7 | simprr 761 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
| 8 | 4, 1, 5, 2, 7 | dchrf 25535 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ) |
| 9 | simprl 759 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
| 10 | 8, 9 | ffvelrnd 6675 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → (𝑥‘𝑎) ∈ ℂ) |
| 11 | 3, 6, 10 | fsumcom 14988 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = Σ𝑥 ∈ 𝐷 Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎)) |
| 12 | eqid 2771 | . . . . . . 7 ⊢ (1r‘(ℤ/nℤ‘𝑁)) = (1r‘(ℤ/nℤ‘𝑁)) | |
| 13 | simpl 475 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑁 ∈ ℕ) | |
| 14 | simpr 477 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
| 15 | 4, 5, 1, 12, 2, 13, 14 | sumdchr2 25563 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 = (1r‘(ℤ/nℤ‘𝑁)), (♯‘𝐷), 0)) |
| 16 | velsn 4451 | . . . . . . 7 ⊢ (𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} ↔ 𝑎 = (1r‘(ℤ/nℤ‘𝑁))) | |
| 17 | ifbi 4365 | . . . . . . 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 2810 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (♯‘𝐷), 0)) |
| 20 | 19 | sumeq2dv 14918 | . . . 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 2771 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 22 | simpr 477 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
| 23 | 4, 1, 5, 21, 22, 2 | dchrsum 25562 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 = (0g‘𝐺), (ϕ‘𝑁), 0)) |
| 24 | velsn 4451 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
| 25 | ifbi 4365 | . . . . . . 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 2810 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 28 | 27 | sumeq2dv 14918 | . . . 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 2815 | . . 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 11713 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
| 31 | 1 | zncrng 20408 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing) |
| 32 | crngring 19043 | . . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring) | |
| 33 | 2, 12 | ringidcl 19053 | . . . . . 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 4612 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(1r‘(ℤ/nℤ‘𝑁))} ⊆ (Base‘(ℤ/nℤ‘𝑁))) |
| 36 | hashcl 13530 | . . . . . 6 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ0) | |
| 37 | nn0cn 11716 | . . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ0 → (♯‘𝐷) ∈ ℂ) | |
| 38 | 6, 36, 37 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) ∈ ℂ) |
| 39 | 38 | ralrimivw 3126 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) ∈ ℂ) |
| 40 | 3 | olcd 861 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((Base‘(ℤ/nℤ‘𝑁)) ⊆ (ℤ≥‘0) ∨ (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)) |
| 41 | sumss2 14941 | . . . 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 826 | . . 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 25547 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
| 44 | ablgrp 18683 | . . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 45 | 5, 21 | grpidcl 17931 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐷) |
| 46 | 43, 44, 45 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0g‘𝐺) ∈ 𝐷) |
| 47 | 46 | snssd 4612 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(0g‘𝐺)} ⊆ 𝐷) |
| 48 | phicl 15960 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | |
| 49 | 48 | nncnd 11455 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℂ) |
| 50 | 49 | ralrimivw 3126 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) ∈ ℂ) |
| 51 | 6 | olcd 861 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐷 ⊆ (ℤ≥‘0) ∨ 𝐷 ∈ Fin)) |
| 52 | sumss2 14941 | . . . 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 826 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
| 54 | 29, 42, 53 | 3eqtr4d 2817 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁)) |
| 55 | eqidd 2772 | . . . 4 ⊢ (𝑎 = (1r‘(ℤ/nℤ‘𝑁)) → (♯‘𝐷) = (♯‘𝐷)) | |
| 56 | 55 | sumsn 14959 | . . 3 ⊢ (((1r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (♯‘𝐷) ∈ ℂ) → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 57 | 34, 38, 56 | syl2anc 576 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (♯‘𝐷) = (♯‘𝐷)) |
| 58 | eqidd 2772 | . . . 4 ⊢ (𝑥 = (0g‘𝐺) → (ϕ‘𝑁) = (ϕ‘𝑁)) | |
| 59 | 58 | sumsn 14959 | . . 3 ⊢ (((0g‘𝐺) ∈ 𝐷 ∧ (ϕ‘𝑁) ∈ ℂ) → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 60 | 46, 49, 59 | syl2anc 576 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
| 61 | 54, 57, 60 | 3eqtr3d 2815 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) = (ϕ‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 834 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ⊆ wss 3822 ifcif 4344 {csn 4435 ‘cfv 6185 Fincfn 8304 ℂcc 10331 0cc0 10333 ℕcn 11437 ℕ0cn0 11705 ℤ≥cuz 12056 ♯chash 13503 Σcsu 14901 ϕcphi 15955 Basecbs 16337 0gc0g 16567 Grpcgrp 17903 Abelcabl 18679 1rcur 18986 Ringcrg 19032 CRingccrg 19033 ℤ/nℤczn 20367 DChrcdchr 25525 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 |
| This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-disj 4894 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-rpss 7265 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-omul 7908 df-er 8087 df-ec 8089 df-qs 8093 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-dju 9122 df-card 9160 df-acn 9163 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-xnn0 11778 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ioc 12557 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-mod 13051 df-seq 13183 df-exp 13243 df-fac 13447 df-bc 13476 df-hash 13504 df-word 13671 df-concat 13732 df-s1 13757 df-shft 14285 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-limsup 14687 df-clim 14704 df-rlim 14705 df-sum 14902 df-ef 15279 df-sin 15281 df-cos 15282 df-pi 15284 df-dvds 15466 df-gcd 15702 df-prm 15870 df-phi 15957 df-pc 16028 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-qus 16636 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-nsg 18073 df-eqg 18074 df-ghm 18139 df-gim 18182 df-ga 18203 df-cntz 18230 df-oppg 18257 df-od 18430 df-gex 18431 df-pgp 18432 df-lsm 18534 df-pj1 18535 df-cmn 18680 df-abl 18681 df-cyg 18765 df-dprd 18879 df-dpj 18880 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-rnghom 19202 df-subrg 19268 df-lmod 19370 df-lss 19438 df-lsp 19478 df-sra 19678 df-rgmod 19679 df-lidl 19680 df-rsp 19681 df-2idl 19738 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-zring 20335 df-zrh 20368 df-zn 20371 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-lp 21463 df-perf 21464 df-cn 21554 df-cnp 21555 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-xms 22648 df-ms 22649 df-tms 22650 df-cncf 23204 df-0p 23989 df-limc 24182 df-dv 24183 df-ply 24496 df-idp 24497 df-coe 24498 df-dgr 24499 df-quot 24598 df-log 24856 df-cxp 24857 df-dchr 25526 |
| This theorem is referenced by: sumdchr 25565 |
| Copyright terms: Public domain | W3C validator |