Step |
Hyp |
Ref |
Expression |
1 |
|
vonsn.1 |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
vonsn.2 |
⊢ ( 𝜑 → 𝐴 ∈ ( ℝ ↑m 𝑋 ) ) |
3 |
|
fveq2 |
⊢ ( 𝑋 = ∅ → ( voln ‘ 𝑋 ) = ( voln ‘ ∅ ) ) |
4 |
3
|
fveq1d |
⊢ ( 𝑋 = ∅ → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = ( ( voln ‘ ∅ ) ‘ { 𝐴 } ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = ( ( voln ‘ ∅ ) ‘ { 𝐴 } ) ) |
6 |
|
0fin |
⊢ ∅ ∈ Fin |
7 |
6
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ∅ ∈ Fin ) |
8 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → 𝐴 ∈ ( ℝ ↑m 𝑋 ) ) |
9 |
|
oveq2 |
⊢ ( 𝑋 = ∅ → ( ℝ ↑m 𝑋 ) = ( ℝ ↑m ∅ ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ℝ ↑m 𝑋 ) = ( ℝ ↑m ∅ ) ) |
11 |
8 10
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → 𝐴 ∈ ( ℝ ↑m ∅ ) ) |
12 |
7 11
|
snvonmbl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → { 𝐴 } ∈ dom ( voln ‘ ∅ ) ) |
13 |
12
|
von0val |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln ‘ ∅ ) ‘ { 𝐴 } ) = 0 ) |
14 |
5 13
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = 0 ) |
15 |
|
neqne |
⊢ ( ¬ 𝑋 = ∅ → 𝑋 ≠ ∅ ) |
16 |
15
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → 𝑋 ≠ ∅ ) |
17 |
2
|
rrxsnicc |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) = { 𝐴 } ) |
18 |
17
|
eqcomd |
⊢ ( 𝜑 → { 𝐴 } = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝜑 → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) ) |
21 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ∈ Fin ) |
22 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ≠ ∅ ) |
23 |
|
elmapi |
⊢ ( 𝐴 ∈ ( ℝ ↑m 𝑋 ) → 𝐴 : 𝑋 ⟶ ℝ ) |
24 |
2 23
|
syl |
⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝐴 : 𝑋 ⟶ ℝ ) |
26 |
|
eqid |
⊢ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) |
27 |
21 22 25 25 26
|
vonn0icc |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) ) |
28 |
24
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ ) |
29 |
28
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ* ) |
30 |
|
iccid |
⊢ ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ* → ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) = { ( 𝐴 ‘ 𝑘 ) } ) |
31 |
29 30
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) = { ( 𝐴 ‘ 𝑘 ) } ) |
32 |
31
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) = ( vol ‘ { ( 𝐴 ‘ 𝑘 ) } ) ) |
33 |
|
volsn |
⊢ ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ → ( vol ‘ { ( 𝐴 ‘ 𝑘 ) } ) = 0 ) |
34 |
28 33
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ { ( 𝐴 ‘ 𝑘 ) } ) = 0 ) |
35 |
32 34
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) = 0 ) |
36 |
35
|
prodeq2dv |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑋 0 ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑋 0 ) |
38 |
|
0cnd |
⊢ ( 𝜑 → 0 ∈ ℂ ) |
39 |
|
fprodconst |
⊢ ( ( 𝑋 ∈ Fin ∧ 0 ∈ ℂ ) → ∏ 𝑘 ∈ 𝑋 0 = ( 0 ↑ ( ♯ ‘ 𝑋 ) ) ) |
40 |
1 38 39
|
syl2anc |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑋 0 = ( 0 ↑ ( ♯ ‘ 𝑋 ) ) ) |
41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑋 0 = ( 0 ↑ ( ♯ ‘ 𝑋 ) ) ) |
42 |
|
hashnncl |
⊢ ( 𝑋 ∈ Fin → ( ( ♯ ‘ 𝑋 ) ∈ ℕ ↔ 𝑋 ≠ ∅ ) ) |
43 |
1 42
|
syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝑋 ) ∈ ℕ ↔ 𝑋 ≠ ∅ ) ) |
44 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( ♯ ‘ 𝑋 ) ∈ ℕ ↔ 𝑋 ≠ ∅ ) ) |
45 |
22 44
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ♯ ‘ 𝑋 ) ∈ ℕ ) |
46 |
|
0exp |
⊢ ( ( ♯ ‘ 𝑋 ) ∈ ℕ → ( 0 ↑ ( ♯ ‘ 𝑋 ) ) = 0 ) |
47 |
45 46
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 0 ↑ ( ♯ ‘ 𝑋 ) ) = 0 ) |
48 |
37 41 47
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,] ( 𝐴 ‘ 𝑘 ) ) ) = 0 ) |
49 |
20 27 48
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = 0 ) |
50 |
16 49
|
syldan |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = 0 ) |
51 |
14 50
|
pm2.61dan |
⊢ ( 𝜑 → ( ( voln ‘ 𝑋 ) ‘ { 𝐴 } ) = 0 ) |