Step |
Hyp |
Ref |
Expression |
1 |
|
hoidmvval0.p |
⊢ Ⅎ 𝑗 𝜑 |
2 |
|
hoidmvval0.l |
⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) |
3 |
|
hoidmvval0.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
4 |
|
hoidmvval0.a |
⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ ) |
5 |
|
hoidmvval0.b |
⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ ) |
6 |
|
hoidmvval0.j |
⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑋 ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) |
7 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
8 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑗 ) ) |
9 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑗 ) ) |
10 |
8 9
|
breq12d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐵 ‘ 𝑘 ) ≤ ( 𝐴 ‘ 𝑘 ) ↔ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) ) |
11 |
10
|
cbvrexvw |
⊢ ( ∃ 𝑘 ∈ 𝑋 ( 𝐵 ‘ 𝑘 ) ≤ ( 𝐴 ‘ 𝑘 ) ↔ ∃ 𝑗 ∈ 𝑋 ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) |
12 |
|
rexn0 |
⊢ ( ∃ 𝑘 ∈ 𝑋 ( 𝐵 ‘ 𝑘 ) ≤ ( 𝐴 ‘ 𝑘 ) → 𝑋 ≠ ∅ ) |
13 |
11 12
|
sylbir |
⊢ ( ∃ 𝑗 ∈ 𝑋 ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) → 𝑋 ≠ ∅ ) |
14 |
6 13
|
syl |
⊢ ( 𝜑 → 𝑋 ≠ ∅ ) |
15 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ∈ Fin ) |
16 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ≠ ∅ ) |
17 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝐴 : 𝑋 ⟶ ℝ ) |
18 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝐵 : 𝑋 ⟶ ℝ ) |
19 |
2 15 16 17 18
|
hoidmvn0val |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
20 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ∃ 𝑗 ∈ 𝑋 ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) |
21 |
|
nfv |
⊢ Ⅎ 𝑗 𝑋 ≠ ∅ |
22 |
1 21
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑋 ≠ ∅ ) |
23 |
|
nfv |
⊢ Ⅎ 𝑗 ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = 0 |
24 |
|
nfv |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) |
25 |
|
nfcv |
⊢ Ⅎ 𝑘 ( vol ‘ ( ( 𝐴 ‘ 𝑗 ) [,) ( 𝐵 ‘ 𝑗 ) ) ) |
26 |
3
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → 𝑋 ∈ Fin ) |
27 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ ) |
28 |
5
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
29 |
|
volicore |
⊢ ( ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∈ ℝ ) |
30 |
27 28 29
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∈ ℝ ) |
31 |
30
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∈ ℂ ) |
32 |
31
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∈ ℂ ) |
33 |
9 8
|
oveq12d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑗 ) [,) ( 𝐵 ‘ 𝑗 ) ) ) |
34 |
33
|
fveq2d |
⊢ ( 𝑘 = 𝑗 → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑗 ) [,) ( 𝐵 ‘ 𝑗 ) ) ) ) |
35 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → 𝑗 ∈ 𝑋 ) |
36 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑗 ) ∈ ℝ ) |
37 |
36
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ( 𝐴 ‘ 𝑗 ) ∈ ℝ ) |
38 |
5
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑗 ) ∈ ℝ ) |
39 |
38
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ( 𝐵 ‘ 𝑗 ) ∈ ℝ ) |
40 |
|
volico |
⊢ ( ( ( 𝐴 ‘ 𝑗 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑗 ) ∈ ℝ ) → ( vol ‘ ( ( 𝐴 ‘ 𝑗 ) [,) ( 𝐵 ‘ 𝑗 ) ) ) = if ( ( 𝐴 ‘ 𝑗 ) < ( 𝐵 ‘ 𝑗 ) , ( ( 𝐵 ‘ 𝑗 ) − ( 𝐴 ‘ 𝑗 ) ) , 0 ) ) |
41 |
37 39 40
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ( vol ‘ ( ( 𝐴 ‘ 𝑗 ) [,) ( 𝐵 ‘ 𝑗 ) ) ) = if ( ( 𝐴 ‘ 𝑗 ) < ( 𝐵 ‘ 𝑗 ) , ( ( 𝐵 ‘ 𝑗 ) − ( 𝐴 ‘ 𝑗 ) ) , 0 ) ) |
42 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) |
43 |
39 37
|
lenltd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ( ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ↔ ¬ ( 𝐴 ‘ 𝑗 ) < ( 𝐵 ‘ 𝑗 ) ) ) |
44 |
42 43
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ¬ ( 𝐴 ‘ 𝑗 ) < ( 𝐵 ‘ 𝑗 ) ) |
45 |
44
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → if ( ( 𝐴 ‘ 𝑗 ) < ( 𝐵 ‘ 𝑗 ) , ( ( 𝐵 ‘ 𝑗 ) − ( 𝐴 ‘ 𝑗 ) ) , 0 ) = 0 ) |
46 |
41 45
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ( vol ‘ ( ( 𝐴 ‘ 𝑗 ) [,) ( 𝐵 ‘ 𝑗 ) ) ) = 0 ) |
47 |
24 25 26 32 34 35 46
|
fprod0 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = 0 ) |
48 |
47
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) ∧ 𝑗 ∈ 𝑋 ∧ ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = 0 ) |
49 |
48
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝑗 ∈ 𝑋 → ( ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = 0 ) ) ) |
50 |
22 23 49
|
rexlimd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ∃ 𝑗 ∈ 𝑋 ( 𝐵 ‘ 𝑗 ) ≤ ( 𝐴 ‘ 𝑗 ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = 0 ) ) |
51 |
20 50
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = 0 ) |
52 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 0 = 0 ) |
53 |
19 51 52
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = 0 ) |
54 |
7 14 53
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = 0 ) |