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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 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋) | |
4 | vonn0ioo2.k | . . . . . . . . . . 11 ⊢ Ⅎ𝑘𝜑 | |
5 | nfv 1910 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑋 | |
6 | 4, 5 | nfan 1895 | . . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋) |
7 | nfcsb1v 3917 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 | |
8 | nfcv 2892 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘ℝ | |
9 | 7, 8 | nfel 2907 | . . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ |
10 | 6, 9 | nfim 1892 | . . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) |
11 | eleq1w 2809 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋)) | |
12 | 11 | anbi2d 628 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝑋))) |
13 | csbeq1a 3906 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) | |
14 | 13 | eleq1d 2811 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)) |
15 | 12, 14 | imbi12d 343 | . . . . . . . . 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 2229 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) |
18 | eqid 2726 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐴) = (𝑘 ∈ 𝑋 ↦ 𝐴) | |
19 | 18 | fvmpts 7012 | . . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
20 | 3, 17, 19 | syl2anc 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
21 | nfcsb1v 3917 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
22 | 21, 8 | nfel 2907 | . . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ |
23 | 6, 22 | nfim 1892 | . . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) |
24 | csbeq1a 3906 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
25 | 24 | eleq1d 2811 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)) |
26 | 12, 25 | imbi12d 343 | . . . . . . . . 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 2229 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) |
29 | eqid 2726 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐵) = (𝑘 ∈ 𝑋 ↦ 𝐵) | |
30 | 29 | fvmpts 7012 | . . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
31 | 3, 28, 30 | syl2anc 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
32 | 20, 31 | oveq12d 7442 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) |
33 | 32 | ixpeq2dva 8941 | . . . . 5 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) |
34 | nfcv 2892 | . . . . . . . 8 ⊢ Ⅎ𝑘(,) | |
35 | 7, 34, 21 | nfov 7454 | . . . . . . 7 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵) |
36 | nfcv 2892 | . . . . . . 7 ⊢ Ⅎ𝑗(𝐴(,)𝐵) | |
37 | 13 | equcoms 2016 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) |
38 | 37 | eqcomd 2732 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴) |
39 | eqidd 2727 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐴 = 𝐴) | |
40 | 38, 39 | eqtrd 2766 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴) |
41 | 24 | equcoms 2016 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) |
42 | 41 | eqcomd 2732 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵) |
43 | 40, 42 | oveq12d 7442 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵) = (𝐴(,)𝐵)) |
44 | 35, 36, 43 | cbvixp 8943 | . . . . . 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 2766 | . . . 4 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑘 ∈ 𝑋 (𝐴(,)𝐵)) |
47 | 2, 46 | eqtr4d 2769 | . . 3 ⊢ (𝜑 → 𝐼 = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)(,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) |
48 | 47 | fveq2d 6905 | . 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 7131 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℝ) |
52 | 4, 27, 29 | fmptdf 7131 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ) |
53 | eqid 2726 | . . 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 46308 | . 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 7442 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)) |
56 | 55 | fveq2d 6905 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = (vol‘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))) |
57 | 17, 28 | voliooico 45613 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))) |
58 | 57 | eqcomd 2732 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵))) |
59 | 56, 58 | eqtrd 2766 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵))) |
60 | 59 | prodeq2dv 15925 | . . 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 6905 | . . . . 5 ⊢ (𝑗 = 𝑘 → (vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(𝐴(,)𝐵))) |
62 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑘𝑋 | |
63 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑗𝑋 | |
64 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑘vol | |
65 | 64, 35 | nffv 6911 | . . . . 5 ⊢ Ⅎ𝑘(vol‘(⦋𝑗 / 𝑘⦌𝐴(,)⦋𝑗 / 𝑘⦌𝐵)) |
66 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑗(vol‘(𝐴(,)𝐵)) | |
67 | 61, 62, 63, 65, 66 | cbvprod 15917 | . . . 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 2766 | . 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵))) |
70 | 48, 54, 69 | 3eqtrd 2770 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴(,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ≠ wne 2930 ⦋csb 3892 ∅c0 4325 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 Xcixp 8926 Fincfn 8974 ℝcr 11157 (,)cioo 13378 [,)cico 13380 ∏cprod 15907 volcvol 25483 volncvoln 46159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cc 10478 ax-ac2 10506 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-disj 5119 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-omul 8501 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-acn 9985 df-ac 10159 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 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-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491 df-sum 15691 df-prod 15908 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-pws 17464 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-rhm 20454 df-subrng 20528 df-subrg 20553 df-drng 20709 df-field 20710 df-abv 20788 df-staf 20818 df-srng 20819 df-lmod 20838 df-lss 20909 df-lmhm 21000 df-lvec 21081 df-sra 21151 df-rgmod 21152 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-cnfld 21344 df-refld 21601 df-phl 21622 df-dsmm 21730 df-frlm 21745 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cn 23222 df-cnp 23223 df-cmp 23382 df-tx 23557 df-hmeo 23750 df-xms 24317 df-ms 24318 df-tms 24319 df-nm 24582 df-ngp 24583 df-tng 24584 df-nrg 24585 df-nlm 24586 df-cncf 24889 df-clm 25081 df-cph 25187 df-tcph 25188 df-rrx 25404 df-ovol 25484 df-vol 25485 df-salg 45930 df-sumge0 45984 df-mea 46071 df-ome 46111 df-caragen 46113 df-ovoln 46158 df-voln 46160 |
This theorem is referenced by: (None) |
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