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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 2610 | . . . . . 6 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | |
2 | eqid 2610 | . . . . . 6 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁)) | |
3 | 1, 2 | znfi 19727 | . . . . 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 24780 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) |
7 | simprr 792 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
8 | 4, 1, 5, 2, 7 | dchrf 24767 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑥:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ) |
9 | simprl 790 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
10 | 8, 9 | ffvelrnd 6268 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ 𝑥 ∈ 𝐷)) → (𝑥‘𝑎) ∈ ℂ) |
11 | 3, 6, 10 | fsumcom 14349 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = Σ𝑥 ∈ 𝐷 Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎)) |
12 | eqid 2610 | . . . . . . 7 ⊢ (1r‘(ℤ/nℤ‘𝑁)) = (1r‘(ℤ/nℤ‘𝑁)) | |
13 | simpl 472 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑁 ∈ ℕ) | |
14 | simpr 476 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) | |
15 | 4, 5, 1, 12, 2, 13, 14 | sumdchr2 24795 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 = (1r‘(ℤ/nℤ‘𝑁)), (#‘𝐷), 0)) |
16 | velsn 4141 | . . . . . . 7 ⊢ (𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} ↔ 𝑎 = (1r‘(ℤ/nℤ‘𝑁))) | |
17 | ifbi 4057 | . . . . . . 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 2647 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))) → Σ𝑥 ∈ 𝐷 (𝑥‘𝑎) = if(𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))}, (#‘𝐷), 0)) |
20 | 19 | sumeq2dv 14281 | . . . 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 2610 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
22 | simpr 476 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
23 | 4, 1, 5, 21, 22, 2 | dchrsum 24794 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 = (0g‘𝐺), (ϕ‘𝑁), 0)) |
24 | velsn 4141 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
25 | ifbi 4057 | . . . . . . 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 2647 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐷) → Σ𝑎 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑥‘𝑎) = if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
28 | 27 | sumeq2dv 14281 | . . . 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 2652 | . . 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 11176 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
31 | 1 | zncrng 19712 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) ∈ CRing) |
32 | crngring 18381 | . . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring) | |
33 | 2, 12 | ringidcl 18391 | . . . . . 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 4281 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(1r‘(ℤ/nℤ‘𝑁))} ⊆ (Base‘(ℤ/nℤ‘𝑁))) |
36 | hashcl 13009 | . . . . . 6 ⊢ (𝐷 ∈ Fin → (#‘𝐷) ∈ ℕ0) | |
37 | nn0cn 11179 | . . . . . 6 ⊢ ((#‘𝐷) ∈ ℕ0 → (#‘𝐷) ∈ ℂ) | |
38 | 6, 36, 37 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (#‘𝐷) ∈ ℂ) |
39 | 38 | ralrimivw 2950 | . . . 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 14304 | . . . 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 1317 | . . 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 24779 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
44 | ablgrp 18021 | . . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
45 | 5, 21 | grpidcl 17273 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐷) |
46 | 43, 44, 45 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0g‘𝐺) ∈ 𝐷) |
47 | 46 | snssd 4281 | . . . 4 ⊢ (𝑁 ∈ ℕ → {(0g‘𝐺)} ⊆ 𝐷) |
48 | phicl 15312 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | |
49 | 48 | nncnd 10913 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℂ) |
50 | 49 | ralrimivw 2950 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) ∈ ℂ) |
51 | 6 | olcd 407 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐷 ⊆ (ℤ≥‘0) ∨ 𝐷 ∈ Fin)) |
52 | sumss2 14304 | . . . 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 1317 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = Σ𝑥 ∈ 𝐷 if(𝑥 ∈ {(0g‘𝐺)}, (ϕ‘𝑁), 0)) |
54 | 29, 42, 53 | 3eqtr4d 2654 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (#‘𝐷) = Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁)) |
55 | eqidd 2611 | . . . 4 ⊢ (𝑎 = (1r‘(ℤ/nℤ‘𝑁)) → (#‘𝐷) = (#‘𝐷)) | |
56 | 55 | sumsn 14319 | . . 3 ⊢ (((1r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (#‘𝐷) ∈ ℂ) → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (#‘𝐷) = (#‘𝐷)) |
57 | 34, 38, 56 | syl2anc 691 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑎 ∈ {(1r‘(ℤ/nℤ‘𝑁))} (#‘𝐷) = (#‘𝐷)) |
58 | eqidd 2611 | . . . 4 ⊢ (𝑥 = (0g‘𝐺) → (ϕ‘𝑁) = (ϕ‘𝑁)) | |
59 | 58 | sumsn 14319 | . . 3 ⊢ (((0g‘𝐺) ∈ 𝐷 ∧ (ϕ‘𝑁) ∈ ℂ) → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
60 | 46, 49, 59 | syl2anc 691 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ {(0g‘𝐺)} (ϕ‘𝑁) = (ϕ‘𝑁)) |
61 | 54, 57, 60 | 3eqtr3d 2652 | 1 ⊢ (𝑁 ∈ ℕ → (#‘𝐷) = (ϕ‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ifcif 4036 {csn 4125 ‘cfv 5804 Fincfn 7841 ℂcc 9813 0cc0 9815 ℕcn 10897 ℕ0cn0 11169 ℤ≥cuz 11563 #chash 12979 Σcsu 14264 ϕcphi 15307 Basecbs 15695 0gc0g 15923 Grpcgrp 17245 Abelcabl 18017 1rcur 18324 Ringcrg 18370 CRingccrg 18371 ℤ/nℤczn 19670 DChrcdchr 24757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-rpss 6835 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-dvds 14822 df-gcd 15055 df-prm 15224 df-phi 15309 df-pc 15380 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-qus 15992 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-gim 17524 df-ga 17546 df-cntz 17573 df-oppg 17599 df-od 17771 df-gex 17772 df-pgp 17773 df-lsm 17874 df-pj1 17875 df-cmn 18018 df-abl 18019 df-cyg 18103 df-dprd 18217 df-dpj 18218 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-0p 23243 df-limc 23436 df-dv 23437 df-ply 23748 df-idp 23749 df-coe 23750 df-dgr 23751 df-quot 23850 df-log 24107 df-cxp 24108 df-dchr 24758 |
This theorem is referenced by: sumdchr 24797 |
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