Step |
Hyp |
Ref |
Expression |
1 |
|
vonhoire.n |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
vonhoire.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
3 |
|
vonhoire.a |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐴 ∈ ℝ ) |
4 |
|
vonhoire.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐵 ∈ ℝ ) |
5 |
|
fveq2 |
⊢ ( 𝑋 = ∅ → ( voln ‘ 𝑋 ) = ( voln ‘ ∅ ) ) |
6 |
5
|
fveq1d |
⊢ ( 𝑋 = ∅ → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) = ( ( voln ‘ ∅ ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) = ( ( voln ‘ ∅ ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) ) |
8 |
|
ixpeq1 |
⊢ ( 𝑋 = ∅ → X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) = X 𝑘 ∈ ∅ ( 𝐴 [,) 𝐵 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) = X 𝑘 ∈ ∅ ( 𝐴 [,) 𝐵 ) ) |
10 |
|
0fin |
⊢ ∅ ∈ Fin |
11 |
10
|
a1i |
⊢ ( 𝜑 → ∅ ∈ Fin ) |
12 |
|
eqid |
⊢ dom ( voln ‘ ∅ ) = dom ( voln ‘ ∅ ) |
13 |
|
noel |
⊢ ¬ 𝑘 ∈ ∅ |
14 |
13
|
pm2.21i |
⊢ ( 𝑘 ∈ ∅ → 𝐴 ∈ ℝ ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ∅ ) → 𝐴 ∈ ℝ ) |
16 |
13
|
pm2.21i |
⊢ ( 𝑘 ∈ ∅ → 𝐵 ∈ ℝ ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ∅ ) → 𝐵 ∈ ℝ ) |
18 |
1 11 12 15 17
|
hoimbl2 |
⊢ ( 𝜑 → X 𝑘 ∈ ∅ ( 𝐴 [,) 𝐵 ) ∈ dom ( voln ‘ ∅ ) ) |
19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → X 𝑘 ∈ ∅ ( 𝐴 [,) 𝐵 ) ∈ dom ( voln ‘ ∅ ) ) |
20 |
9 19
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ∈ dom ( voln ‘ ∅ ) ) |
21 |
20
|
von0val |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln ‘ ∅ ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) = 0 ) |
22 |
7 21
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) = 0 ) |
23 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → 0 ∈ ℝ ) |
24 |
22 23
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) ∈ ℝ ) |
25 |
|
neqne |
⊢ ( ¬ 𝑋 = ∅ → 𝑋 ≠ ∅ ) |
26 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → 𝑋 ≠ ∅ ) |
27 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → 𝑗 ∈ 𝑋 ) |
28 |
|
nfv |
⊢ Ⅎ 𝑘 𝑗 ∈ 𝑋 |
29 |
1 28
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) |
30 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑗 |
31 |
30
|
nfcsb1 |
⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐴 |
32 |
31
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ∈ ℝ |
33 |
29 32
|
nfim |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ∈ ℝ ) |
34 |
|
eleq1w |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋 ) ) |
35 |
34
|
anbi2d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) ) ) |
36 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑗 → 𝐴 = ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ) |
37 |
36
|
eleq1d |
⊢ ( 𝑘 = 𝑗 → ( 𝐴 ∈ ℝ ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ∈ ℝ ) ) |
38 |
35 37
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐴 ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ∈ ℝ ) ) ) |
39 |
33 38 3
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ∈ ℝ ) |
40 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) = ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) |
41 |
30 31 36 40
|
fvmptf |
⊢ ( ( 𝑗 ∈ 𝑋 ∧ ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ∈ ℝ ) → ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ) |
42 |
27 39 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐴 ) |
43 |
30
|
nfcsb1 |
⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 |
44 |
|
nfcv |
⊢ Ⅎ 𝑘 ℝ |
45 |
43 44
|
nfel |
⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℝ |
46 |
29 45
|
nfim |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℝ ) |
47 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑗 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) |
48 |
47
|
eleq1d |
⊢ ( 𝑘 = 𝑗 → ( 𝐵 ∈ ℝ ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℝ ) ) |
49 |
35 48
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐵 ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℝ ) ) ) |
50 |
46 49 4
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℝ ) |
51 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) = ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) |
52 |
30 43 47 51
|
fvmptf |
⊢ ( ( 𝑗 ∈ 𝑋 ∧ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℝ ) → ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) |
53 |
27 50 52
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) |
54 |
42 53
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) = ( ⦋ 𝑗 / 𝑘 ⦌ 𝐴 [,) ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) ) |
55 |
54
|
ixpeq2dva |
⊢ ( 𝜑 → X 𝑗 ∈ 𝑋 ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) = X 𝑗 ∈ 𝑋 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐴 [,) ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) ) |
56 |
|
nfcv |
⊢ Ⅎ 𝑗 ( 𝐴 [,) 𝐵 ) |
57 |
|
nfcv |
⊢ Ⅎ 𝑘 [,) |
58 |
31 57 43
|
nfov |
⊢ Ⅎ 𝑘 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐴 [,) ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) |
59 |
36 47
|
oveq12d |
⊢ ( 𝑘 = 𝑗 → ( 𝐴 [,) 𝐵 ) = ( ⦋ 𝑗 / 𝑘 ⦌ 𝐴 [,) ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) ) |
60 |
56 58 59
|
cbvixp |
⊢ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) = X 𝑗 ∈ 𝑋 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐴 [,) ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) |
61 |
60
|
eqcomi |
⊢ X 𝑗 ∈ 𝑋 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐴 [,) ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) = X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) |
62 |
61
|
a1i |
⊢ ( 𝜑 → X 𝑗 ∈ 𝑋 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐴 [,) ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) = X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) |
63 |
55 62
|
eqtr2d |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) = X 𝑗 ∈ 𝑋 ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) |
64 |
63
|
fveq2d |
⊢ ( 𝜑 → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) = ( ( voln ‘ 𝑋 ) ‘ X 𝑗 ∈ 𝑋 ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ) |
65 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) = ( ( voln ‘ 𝑋 ) ‘ X 𝑗 ∈ 𝑋 ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ) |
66 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ∈ Fin ) |
67 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ≠ ∅ ) |
68 |
1 3 40
|
fmptdf |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) : 𝑋 ⟶ ℝ ) |
69 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) : 𝑋 ⟶ ℝ ) |
70 |
1 4 51
|
fmptdf |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) : 𝑋 ⟶ ℝ ) |
71 |
70
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) : 𝑋 ⟶ ℝ ) |
72 |
|
eqid |
⊢ X 𝑗 ∈ 𝑋 ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) = X 𝑗 ∈ 𝑋 ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) |
73 |
66 67 69 71 72
|
vonn0hoi |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑗 ∈ 𝑋 ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) = ∏ 𝑗 ∈ 𝑋 ( vol ‘ ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ) |
74 |
65 73
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) = ∏ 𝑗 ∈ 𝑋 ( vol ‘ ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ) |
75 |
42 39
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) ∈ ℝ ) |
76 |
53 50
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ∈ ℝ ) |
77 |
|
volicore |
⊢ ( ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) ∈ ℝ ∧ ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ∈ ℝ ) → ( vol ‘ ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ∈ ℝ ) |
78 |
75 76 77
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( vol ‘ ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ∈ ℝ ) |
79 |
2 78
|
fprodrecl |
⊢ ( 𝜑 → ∏ 𝑗 ∈ 𝑋 ( vol ‘ ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ∈ ℝ ) |
80 |
79
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ∏ 𝑗 ∈ 𝑋 ( vol ‘ ( ( ( 𝑘 ∈ 𝑋 ↦ 𝐴 ) ‘ 𝑗 ) [,) ( ( 𝑘 ∈ 𝑋 ↦ 𝐵 ) ‘ 𝑗 ) ) ) ∈ ℝ ) |
81 |
74 80
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) ∈ ℝ ) |
82 |
26 81
|
syldan |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) ∈ ℝ ) |
83 |
24 82
|
pm2.61dan |
⊢ ( 𝜑 → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ) ∈ ℝ ) |