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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonn0ioo2 | Structured version Visualization version GIF version |
Description: The n-dimensional Lebesgue measure of an open interval when the dimension of the space is nonzero. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
vonn0ioo2.k | ⊢ Ⅎ𝑘𝜑 |
vonn0ioo2.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
vonn0ioo2.n | ⊢ (𝜑 → 𝑋 ≠ ∅) |
vonn0ioo2.a | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) |
vonn0ioo2.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ) |
vonn0ioo2.i | ⊢ 𝐼 = X𝑘 ∈ 𝑋 (𝐴(,)𝐵) |
Ref | Expression |
---|---|
vonn0ioo2 | ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonn0ioo2.i | . . . . 5 ⊢ 𝐼 = X𝑘 ∈ 𝑋 (𝐴(,)𝐵) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 (𝐴(,)𝐵)) |
3 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋) | |
4 | vonn0ioo2.k | . . . . . . . . . . 11 ⊢ Ⅎ𝑘𝜑 | |
5 | nfv 1917 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑋 | |
6 | 4, 5 | nfan 1902 | . . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋) |
7 | nfcsb1v 3880 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 | |
8 | nfcv 2907 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘ℝ | |
9 | 7, 8 | nfel 2921 | . . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ |
10 | 6, 9 | nfim 1899 | . . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) |
11 | eleq1w 2820 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋)) | |
12 | 11 | anbi2d 629 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝑋))) |
13 | csbeq1a 3869 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) | |
14 | 13 | eleq1d 2822 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)) |
15 | 12, 14 | imbi12d 344 | . . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ))) |
16 | vonn0ioo2.a | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) | |
17 | 10, 15, 16 | chvarfv 2233 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) |
18 | eqid 2736 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐴) = (𝑘 ∈ 𝑋 ↦ 𝐴) | |
19 | 18 | fvmpts 6951 | . . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
20 | 3, 17, 19 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
21 | nfcsb1v 3880 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
22 | 21, 8 | nfel 2921 | . . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ |
23 | 6, 22 | nfim 1899 | . . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) |
24 | csbeq1a 3869 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
25 | 24 | eleq1d 2822 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)) |
26 | 12, 25 | imbi12d 344 | . . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ))) |
27 | vonn0ioo2.b | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ) | |
28 | 23, 26, 27 | chvarfv 2233 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) |
29 | eqid 2736 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐵) = (𝑘 ∈ 𝑋 ↦ 𝐵) | |
30 | 29 | fvmpts 6951 | . . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
31 | 3, 28, 30 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
32 | 20, 31 | oveq12d 7375 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) |
33 | 32 | ixpeq2dva 8850 | . . . . 5 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) |
34 | nfcv 2907 | . . . . . . . 8 ⊢ Ⅎ𝑘(,) | |
35 | 7, 34, 21 | nfov 7387 | . . . . . . 7 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵) |
36 | nfcv 2907 | . . . . . . 7 ⊢ Ⅎ𝑗(𝐴(,)𝐵) | |
37 | 13 | equcoms 2023 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) |
38 | 37 | eqcomd 2742 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴) |
39 | eqidd 2737 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐴 = 𝐴) | |
40 | 38, 39 | eqtrd 2776 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴) |
41 | 24 | equcoms 2023 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) |
42 | 41 | eqcomd 2742 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵) |
43 | 40, 42 | oveq12d 7375 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵) = (𝐴(,)𝐵)) |
44 | 35, 36, 43 | cbvixp 8852 | . . . . . 6 ⊢ X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵) = X𝑘 ∈ 𝑋 (𝐴(,)𝐵) |
45 | 44 | a1i 11 | . . . . 5 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵) = X𝑘 ∈ 𝑋 (𝐴(,)𝐵)) |
46 | 33, 45 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑘 ∈ 𝑋 (𝐴(,)𝐵)) |
47 | 2, 46 | eqtr4d 2779 | . . 3 ⊢ (𝜑 → 𝐼 = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) |
48 | 47 | fveq2d 6846 | . 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)))) |
49 | vonn0ioo2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
50 | vonn0ioo2.n | . . 3 ⊢ (𝜑 → 𝑋 ≠ ∅) | |
51 | 4, 16, 18 | fmptdf 7065 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℝ) |
52 | 4, 27, 29 | fmptdf 7065 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ) |
53 | eqid 2736 | . . 3 ⊢ X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) | |
54 | 49, 50, 51, 52, 53 | vonn0ioo 44918 | . 2 ⊢ (𝜑 → ((voln‘𝑋)‘X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)))) |
55 | 20, 31 | oveq12d 7375 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)) |
56 | 55 | fveq2d 6846 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = (vol‘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))) |
57 | 17, 28 | voliooico 44223 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))) |
58 | 57 | eqcomd 2742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵))) |
59 | 56, 58 | eqtrd 2776 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵))) |
60 | 59 | prodeq2dv 15806 | . . 3 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑗 ∈ 𝑋 (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵))) |
61 | 43 | fveq2d 6846 | . . . . 5 ⊢ (𝑗 = 𝑘 → (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(𝐴(,)𝐵))) |
62 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑘𝑋 | |
63 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑗𝑋 | |
64 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑘vol | |
65 | 64, 35 | nffv 6852 | . . . . 5 ⊢ Ⅎ𝑘(vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) |
66 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑗(vol‘(𝐴(,)𝐵)) | |
67 | 61, 62, 63, 65, 66 | cbvprod 15798 | . . . 4 ⊢ ∏𝑗 ∈ 𝑋 (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵)) |
68 | 67 | a1i 11 | . . 3 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵))) |
69 | 60, 68 | eqtrd 2776 | . 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵))) |
70 | 48, 54, 69 | 3eqtrd 2780 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ≠ wne 2943 ⦋csb 3855 ∅c0 4282 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Xcixp 8835 Fincfn 8883 ℝcr 11050 (,)cioo 13264 [,)cico 13266 ∏cprod 15788 volcvol 24827 volncvoln 44769 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-ac2 10399 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-ac 10052 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-prod 15789 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-pws 17331 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-abv 20276 df-staf 20304 df-srng 20305 df-lmod 20324 df-lss 20393 df-lmhm 20483 df-lvec 20564 df-sra 20633 df-rgmod 20634 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-refld 21009 df-phl 21030 df-dsmm 21138 df-frlm 21153 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cn 22578 df-cnp 22579 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675 df-nm 23938 df-ngp 23939 df-tng 23940 df-nrg 23941 df-nlm 23942 df-cncf 24241 df-clm 24426 df-cph 24532 df-tcph 24533 df-rrx 24749 df-ovol 24828 df-vol 24829 df-salg 44540 df-sumge0 44594 df-mea 44681 df-ome 44721 df-caragen 44723 df-ovoln 44768 df-voln 44770 |
This theorem is referenced by: (None) |
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