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Mirrors > Home > MPE Home > Th. List > dchrvmasumlema | Structured version Visualization version GIF version |
Description: Lemma for dchrvmasum 25259 and dchrvmasumif 25237. Apply dchrisum 25226 for the function log(𝑦) / 𝑦, which is decreasing above e (or above 3, the nearest integer bound). (Contributed by Mario Carneiro, 5-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
rpvmasum.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
rpvmasum.a | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
rpvmasum.g | ⊢ 𝐺 = (DChr‘𝑁) |
rpvmasum.d | ⊢ 𝐷 = (Base‘𝐺) |
rpvmasum.1 | ⊢ 1 = (0g‘𝐺) |
dchrisum.b | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrisum.n1 | ⊢ (𝜑 → 𝑋 ≠ 1 ) |
dchrvmasumlema.f | ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) |
Ref | Expression |
---|---|
dchrvmasumlema | ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
2 | rpvmasum.l | . . 3 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
3 | rpvmasum.a | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | rpvmasum.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
5 | rpvmasum.d | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
6 | rpvmasum.1 | . . 3 ⊢ 1 = (0g‘𝐺) | |
7 | dchrisum.b | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
8 | dchrisum.n1 | . . 3 ⊢ (𝜑 → 𝑋 ≠ 1 ) | |
9 | fveq2 6229 | . . . 4 ⊢ (𝑛 = 𝑥 → (log‘𝑛) = (log‘𝑥)) | |
10 | id 22 | . . . 4 ⊢ (𝑛 = 𝑥 → 𝑛 = 𝑥) | |
11 | 9, 10 | oveq12d 6708 | . . 3 ⊢ (𝑛 = 𝑥 → ((log‘𝑛) / 𝑛) = ((log‘𝑥) / 𝑥)) |
12 | 3nn 11224 | . . . 4 ⊢ 3 ∈ ℕ | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → 3 ∈ ℕ) |
14 | relogcl 24367 | . . . . 5 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ) | |
15 | rerpdivcl 11899 | . . . . 5 ⊢ (((log‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ+) → ((log‘𝑛) / 𝑛) ∈ ℝ) | |
16 | 14, 15 | mpancom 704 | . . . 4 ⊢ (𝑛 ∈ ℝ+ → ((log‘𝑛) / 𝑛) ∈ ℝ) |
17 | 16 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ+) → ((log‘𝑛) / 𝑛) ∈ ℝ) |
18 | simp3r 1110 | . . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥) | |
19 | simp2l 1107 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ+) | |
20 | 19 | rpred 11910 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ) |
21 | ere 14863 | . . . . . . 7 ⊢ e ∈ ℝ | |
22 | 21 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ∈ ℝ) |
23 | 3re 11132 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 3 ∈ ℝ) |
25 | egt2lt3 14978 | . . . . . . . . 9 ⊢ (2 < e ∧ e < 3) | |
26 | 25 | simpri 477 | . . . . . . . 8 ⊢ e < 3 |
27 | 21, 23, 26 | ltleii 10198 | . . . . . . 7 ⊢ e ≤ 3 |
28 | 27 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ≤ 3) |
29 | simp3l 1109 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 3 ≤ 𝑛) | |
30 | 22, 24, 20, 28, 29 | letrd 10232 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ≤ 𝑛) |
31 | simp2r 1108 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ+) | |
32 | 31 | rpred 11910 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
33 | 22, 20, 32, 30, 18 | letrd 10232 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ≤ 𝑥) |
34 | logdivle 24413 | . . . . 5 ⊢ (((𝑛 ∈ ℝ ∧ e ≤ 𝑛) ∧ (𝑥 ∈ ℝ ∧ e ≤ 𝑥)) → (𝑛 ≤ 𝑥 ↔ ((log‘𝑥) / 𝑥) ≤ ((log‘𝑛) / 𝑛))) | |
35 | 20, 30, 32, 33, 34 | syl22anc 1367 | . . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑛 ≤ 𝑥 ↔ ((log‘𝑥) / 𝑥) ≤ ((log‘𝑛) / 𝑛))) |
36 | 18, 35 | mpbid 222 | . . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → ((log‘𝑥) / 𝑥) ≤ ((log‘𝑛) / 𝑛)) |
37 | rpcn 11879 | . . . . . . 7 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℂ) | |
38 | 37 | cxp1d 24497 | . . . . . 6 ⊢ (𝑛 ∈ ℝ+ → (𝑛↑𝑐1) = 𝑛) |
39 | 38 | oveq2d 6706 | . . . . 5 ⊢ (𝑛 ∈ ℝ+ → ((log‘𝑛) / (𝑛↑𝑐1)) = ((log‘𝑛) / 𝑛)) |
40 | 39 | mpteq2ia 4773 | . . . 4 ⊢ (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / (𝑛↑𝑐1))) = (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / 𝑛)) |
41 | 1rp 11874 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
42 | cxploglim 24749 | . . . . 5 ⊢ (1 ∈ ℝ+ → (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / (𝑛↑𝑐1))) ⇝𝑟 0) | |
43 | 41, 42 | mp1i 13 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / (𝑛↑𝑐1))) ⇝𝑟 0) |
44 | 40, 43 | syl5eqbrr 4721 | . . 3 ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / 𝑛)) ⇝𝑟 0) |
45 | dchrvmasumlema.f | . . . 4 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) | |
46 | fveq2 6229 | . . . . . . 7 ⊢ (𝑎 = 𝑛 → (𝐿‘𝑎) = (𝐿‘𝑛)) | |
47 | 46 | fveq2d 6233 | . . . . . 6 ⊢ (𝑎 = 𝑛 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑛))) |
48 | fveq2 6229 | . . . . . . 7 ⊢ (𝑎 = 𝑛 → (log‘𝑎) = (log‘𝑛)) | |
49 | id 22 | . . . . . . 7 ⊢ (𝑎 = 𝑛 → 𝑎 = 𝑛) | |
50 | 48, 49 | oveq12d 6708 | . . . . . 6 ⊢ (𝑎 = 𝑛 → ((log‘𝑎) / 𝑎) = ((log‘𝑛) / 𝑛)) |
51 | 47, 50 | oveq12d 6708 | . . . . 5 ⊢ (𝑎 = 𝑛 → ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)) = ((𝑋‘(𝐿‘𝑛)) · ((log‘𝑛) / 𝑛))) |
52 | 51 | cbvmptv 4783 | . . . 4 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · ((log‘𝑛) / 𝑛))) |
53 | 45, 52 | eqtri 2673 | . . 3 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · ((log‘𝑛) / 𝑛))) |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 11, 13, 17, 36, 44, 53 | dchrisum 25226 | . 2 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥)))) |
55 | fveq2 6229 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (⌊‘𝑥) = (⌊‘𝑦)) | |
56 | 55 | fveq2d 6233 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (seq1( + , 𝐹)‘(⌊‘𝑥)) = (seq1( + , 𝐹)‘(⌊‘𝑦))) |
57 | 56 | oveq1d 6705 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡) = ((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) |
58 | 57 | fveq2d 6233 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡))) |
59 | fveq2 6229 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (log‘𝑥) = (log‘𝑦)) | |
60 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
61 | 59, 60 | oveq12d 6708 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((log‘𝑥) / 𝑥) = ((log‘𝑦) / 𝑦)) |
62 | 61 | oveq2d 6706 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑐 · ((log‘𝑥) / 𝑥)) = (𝑐 · ((log‘𝑦) / 𝑦))) |
63 | 58, 62 | breq12d 4698 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥)) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) |
64 | 63 | cbvralv 3201 | . . . . 5 ⊢ (∀𝑥 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥)) ↔ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))) |
65 | 64 | anbi2i 730 | . . . 4 ⊢ ((seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥))) ↔ (seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) |
66 | 65 | rexbii 3070 | . . 3 ⊢ (∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥))) ↔ ∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) |
67 | 66 | exbii 1814 | . 2 ⊢ (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥))) ↔ ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) |
68 | 54, 67 | sylib 208 | 1 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∃wex 1744 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 ∃wrex 2942 class class class wbr 4685 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 +∞cpnf 10109 < clt 10112 ≤ cle 10113 − cmin 10304 / cdiv 10722 ℕcn 11058 2c2 11108 3c3 11109 ℝ+crp 11870 [,)cico 12215 ⌊cfl 12631 seqcseq 12841 abscabs 14018 ⇝ cli 14259 ⇝𝑟 crli 14260 eceu 14837 Basecbs 15904 0gc0g 16147 ℤRHomczrh 19896 ℤ/nℤczn 19899 logclog 24346 ↑𝑐ccxp 24347 DChrcdchr 25002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-ec 7789 df-qs 7793 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-xnn0 11402 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-e 14843 df-sin 14844 df-cos 14845 df-pi 14847 df-dvds 15028 df-gcd 15264 df-phi 15518 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-qus 16216 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-nsg 17639 df-eqg 17640 df-ghm 17705 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-rnghom 18763 df-subrg 18826 df-lmod 18913 df-lss 18981 df-lsp 19020 df-sra 19220 df-rgmod 19221 df-lidl 19222 df-rsp 19223 df-2idl 19280 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-zring 19867 df-zrh 19900 df-zn 19903 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348 df-cxp 24349 df-dchr 25003 |
This theorem is referenced by: dchrvmasumif 25237 |
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