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Mirrors > Home > MPE Home > Th. List > dchrvmasumlema | Structured version Visualization version GIF version |
Description: Lemma for dchrvmasum 27584 and dchrvmasumif 27562. Apply dchrisum 27551 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 6907 | . . . 4 ⊢ (𝑛 = 𝑥 → (log‘𝑛) = (log‘𝑥)) | |
10 | id 22 | . . . 4 ⊢ (𝑛 = 𝑥 → 𝑛 = 𝑥) | |
11 | 9, 10 | oveq12d 7449 | . . 3 ⊢ (𝑛 = 𝑥 → ((log‘𝑛) / 𝑛) = ((log‘𝑥) / 𝑥)) |
12 | 3nn 12343 | . . . 4 ⊢ 3 ∈ ℕ | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → 3 ∈ ℕ) |
14 | relogcl 26632 | . . . . 5 ⊢ (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ) | |
15 | rerpdivcl 13063 | . . . . 5 ⊢ (((log‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ+) → ((log‘𝑛) / 𝑛) ∈ ℝ) | |
16 | 14, 15 | mpancom 688 | . . . 4 ⊢ (𝑛 ∈ ℝ+ → ((log‘𝑛) / 𝑛) ∈ ℝ) |
17 | 16 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ+) → ((log‘𝑛) / 𝑛) ∈ ℝ) |
18 | simp3r 1201 | . . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥) | |
19 | simp2l 1198 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ+) | |
20 | 19 | rpred 13075 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ) |
21 | ere 16122 | . . . . . . 7 ⊢ e ∈ ℝ | |
22 | 21 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ∈ ℝ) |
23 | 3re 12344 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 3 ∈ ℝ) |
25 | egt2lt3 16239 | . . . . . . . . 9 ⊢ (2 < e ∧ e < 3) | |
26 | 25 | simpri 485 | . . . . . . . 8 ⊢ e < 3 |
27 | 21, 23, 26 | ltleii 11382 | . . . . . . 7 ⊢ e ≤ 3 |
28 | 27 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ≤ 3) |
29 | simp3l 1200 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 3 ≤ 𝑛) | |
30 | 22, 24, 20, 28, 29 | letrd 11416 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ≤ 𝑛) |
31 | simp2r 1199 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ+) | |
32 | 31 | rpred 13075 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
33 | 22, 20, 32, 30, 18 | letrd 11416 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → e ≤ 𝑥) |
34 | logdivle 26679 | . . . . 5 ⊢ (((𝑛 ∈ ℝ ∧ e ≤ 𝑛) ∧ (𝑥 ∈ ℝ ∧ e ≤ 𝑥)) → (𝑛 ≤ 𝑥 ↔ ((log‘𝑥) / 𝑥) ≤ ((log‘𝑛) / 𝑛))) | |
35 | 20, 30, 32, 33, 34 | syl22anc 839 | . . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑛 ≤ 𝑥 ↔ ((log‘𝑥) / 𝑥) ≤ ((log‘𝑛) / 𝑛))) |
36 | 18, 35 | mpbid 232 | . . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) ∧ (3 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → ((log‘𝑥) / 𝑥) ≤ ((log‘𝑛) / 𝑛)) |
37 | rpcn 13043 | . . . . . . 7 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℂ) | |
38 | 37 | cxp1d 26763 | . . . . . 6 ⊢ (𝑛 ∈ ℝ+ → (𝑛↑𝑐1) = 𝑛) |
39 | 38 | oveq2d 7447 | . . . . 5 ⊢ (𝑛 ∈ ℝ+ → ((log‘𝑛) / (𝑛↑𝑐1)) = ((log‘𝑛) / 𝑛)) |
40 | 39 | mpteq2ia 5251 | . . . 4 ⊢ (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / (𝑛↑𝑐1))) = (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / 𝑛)) |
41 | 1rp 13036 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
42 | cxploglim 27036 | . . . . 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 | eqbrtrrid 5184 | . . 3 ⊢ (𝜑 → (𝑛 ∈ ℝ+ ↦ ((log‘𝑛) / 𝑛)) ⇝𝑟 0) |
45 | dchrvmasumlema.f | . . . 4 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) | |
46 | 2fveq3 6912 | . . . . . 6 ⊢ (𝑎 = 𝑛 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑛))) | |
47 | fveq2 6907 | . . . . . . 7 ⊢ (𝑎 = 𝑛 → (log‘𝑎) = (log‘𝑛)) | |
48 | id 22 | . . . . . . 7 ⊢ (𝑎 = 𝑛 → 𝑎 = 𝑛) | |
49 | 47, 48 | oveq12d 7449 | . . . . . 6 ⊢ (𝑎 = 𝑛 → ((log‘𝑎) / 𝑎) = ((log‘𝑛) / 𝑛)) |
50 | 46, 49 | oveq12d 7449 | . . . . 5 ⊢ (𝑎 = 𝑛 → ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)) = ((𝑋‘(𝐿‘𝑛)) · ((log‘𝑛) / 𝑛))) |
51 | 50 | cbvmptv 5261 | . . . 4 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · ((log‘𝑛) / 𝑛))) |
52 | 45, 51 | eqtri 2763 | . . 3 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · ((log‘𝑛) / 𝑛))) |
53 | 1, 2, 3, 4, 5, 6, 7, 8, 11, 13, 17, 36, 44, 52 | dchrisum 27551 | . 2 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥)))) |
54 | 2fveq3 6912 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (seq1( + , 𝐹)‘(⌊‘𝑥)) = (seq1( + , 𝐹)‘(⌊‘𝑦))) | |
55 | 54 | fvoveq1d 7453 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡))) |
56 | fveq2 6907 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (log‘𝑥) = (log‘𝑦)) | |
57 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
58 | 56, 57 | oveq12d 7449 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((log‘𝑥) / 𝑥) = ((log‘𝑦) / 𝑦)) |
59 | 58 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑐 · ((log‘𝑥) / 𝑥)) = (𝑐 · ((log‘𝑦) / 𝑦))) |
60 | 55, 59 | breq12d 5161 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥)) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))) |
61 | 60 | cbvralvw 3235 | . . . . 5 ⊢ (∀𝑥 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · ((log‘𝑥) / 𝑥)) ↔ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))) |
62 | 61 | anbi2i 623 | . . . 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)))) |
63 | 62 | rexbii 3092 | . . 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)))) |
64 | 63 | exbii 1845 | . 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)))) |
65 | 53, 64 | sylib 218 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 +∞cpnf 11290 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 3c3 12320 ℝ+crp 13032 [,)cico 13386 ⌊cfl 13827 seqcseq 14039 abscabs 15270 ⇝ cli 15517 ⇝𝑟 crli 15518 eceu 16095 Basecbs 17245 0gc0g 17486 ℤRHomczrh 21528 ℤ/nℤczn 21531 logclog 26611 ↑𝑐ccxp 26612 DChrcdchr 27291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-e 16101 df-sin 16102 df-cos 16103 df-pi 16105 df-dvds 16288 df-gcd 16529 df-phi 16800 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-qus 17556 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614 df-dchr 27292 |
This theorem is referenced by: dchrvmasumif 27562 |
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