Step |
Hyp |
Ref |
Expression |
1 |
|
hoidmv1le.l |
⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) |
2 |
|
hoidmv1le.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
3 |
|
hoidmv1le.x |
⊢ 𝑋 = { 𝑍 } |
4 |
|
hoidmv1le.a |
⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ ) |
5 |
|
hoidmv1le.b |
⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ ) |
6 |
|
hoidmv1le.c |
⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
7 |
|
hoidmv1le.d |
⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
8 |
|
hoidmv1le.s |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
9 |
|
snidg |
⊢ ( 𝑍 ∈ 𝑉 → 𝑍 ∈ { 𝑍 } ) |
10 |
2 9
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ { 𝑍 } ) |
11 |
10 3
|
eleqtrrdi |
⊢ ( 𝜑 → 𝑍 ∈ 𝑋 ) |
12 |
5 11
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ ) |
13 |
4 11
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) |
14 |
12 13
|
resubcld |
⊢ ( 𝜑 → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ ) |
15 |
14
|
rexrd |
⊢ ( 𝜑 → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ* ) |
16 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
17 |
16
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
18 |
14
|
ltpnfd |
⊢ ( 𝜑 → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) < +∞ ) |
19 |
15 17 18
|
xrltled |
⊢ ( 𝜑 → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ +∞ ) |
20 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ +∞ ) |
21 |
|
id |
⊢ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) |
22 |
21
|
eqcomd |
⊢ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
23 |
22
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
24 |
20 23
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
25 |
|
simpl |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ) |
26 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) |
27 |
|
nnex |
⊢ ℕ ∈ V |
28 |
27
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ℕ ∈ V ) |
29 |
3
|
a1i |
⊢ ( 𝜑 → 𝑋 = { 𝑍 } ) |
30 |
|
snfi |
⊢ { 𝑍 } ∈ Fin |
31 |
30
|
a1i |
⊢ ( 𝜑 → { 𝑍 } ∈ Fin ) |
32 |
29 31
|
eqeltrd |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑋 ∈ Fin ) |
34 |
11
|
ne0d |
⊢ ( 𝜑 → 𝑋 ≠ ∅ ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑋 ≠ ∅ ) |
36 |
6
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) ) |
37 |
|
elmapi |
⊢ ( ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) → ( 𝐶 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
38 |
36 37
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
39 |
7
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) ) |
40 |
|
elmapi |
⊢ ( ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) → ( 𝐷 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
41 |
39 40
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
42 |
1 33 35 38 41
|
hoidmvn0val |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
43 |
3
|
prodeq1i |
⊢ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
44 |
43
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
45 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ 𝑉 ) |
46 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ 𝑋 ) |
47 |
38 46
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
48 |
41 46
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
49 |
|
volicore |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ∈ ℝ ) |
50 |
47 48 49
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ∈ ℝ ) |
51 |
50
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ∈ ℂ ) |
52 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) |
53 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
54 |
52 53
|
oveq12d |
⊢ ( 𝑘 = 𝑍 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
55 |
54
|
fveq2d |
⊢ ( 𝑘 = 𝑍 → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
56 |
55
|
prodsn |
⊢ ( ( 𝑍 ∈ 𝑉 ∧ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ∈ ℂ ) → ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
57 |
45 51 56
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
58 |
42 44 57
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
59 |
58
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) |
60 |
|
fveq2 |
⊢ ( 𝑘 = 𝑙 → ( 𝑎 ‘ 𝑘 ) = ( 𝑎 ‘ 𝑙 ) ) |
61 |
|
fveq2 |
⊢ ( 𝑘 = 𝑙 → ( 𝑏 ‘ 𝑘 ) = ( 𝑏 ‘ 𝑙 ) ) |
62 |
60 61
|
oveq12d |
⊢ ( 𝑘 = 𝑙 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) |
63 |
62
|
fveq2d |
⊢ ( 𝑘 = 𝑙 → ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ) |
64 |
63
|
cbvprodv |
⊢ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) |
65 |
|
ifeq2 |
⊢ ( ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ) ) |
66 |
64 65
|
ax-mp |
⊢ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ) |
67 |
66
|
a1i |
⊢ ( ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) ∧ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ) → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ) ) |
68 |
67
|
mpoeq3ia |
⊢ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) = ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ) ) |
69 |
68
|
mpteq2i |
⊢ ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ) ) ) |
70 |
1 69
|
eqtri |
⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ) ) ) |
71 |
70 33 38 41
|
hoidmvcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) ) |
72 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) |
73 |
71 72
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) : ℕ ⟶ ( 0 [,) +∞ ) ) |
74 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
75 |
74
|
a1i |
⊢ ( 𝜑 → ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) ) |
76 |
73 75
|
fssd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
77 |
59 76
|
feq1dd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
78 |
77
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
79 |
28 78
|
sge0repnf |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ↔ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) ) |
80 |
26 79
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) |
81 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) |
82 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( 𝐵 ‘ 𝑍 ) ∈ ℝ ) |
83 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) |
84 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) |
85 |
47 84
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) : ℕ ⟶ ℝ ) |
86 |
85
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) : ℕ ⟶ ℝ ) |
87 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
88 |
48 87
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) : ℕ ⟶ ℝ ) |
89 |
88
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) : ℕ ⟶ ℝ ) |
90 |
3
|
eleq2i |
⊢ ( 𝑘 ∈ 𝑋 ↔ 𝑘 ∈ { 𝑍 } ) |
91 |
90
|
biimpi |
⊢ ( 𝑘 ∈ 𝑋 → 𝑘 ∈ { 𝑍 } ) |
92 |
|
elsni |
⊢ ( 𝑘 ∈ { 𝑍 } → 𝑘 = 𝑍 ) |
93 |
91 92
|
syl |
⊢ ( 𝑘 ∈ 𝑋 → 𝑘 = 𝑍 ) |
94 |
93 54
|
syl |
⊢ ( 𝑘 ∈ 𝑋 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
95 |
94
|
rgen |
⊢ ∀ 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
96 |
|
ixpeq2 |
⊢ ( ∀ 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
97 |
95 96
|
ax-mp |
⊢ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
98 |
97
|
a1i |
⊢ ( 𝑗 ∈ ℕ → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
99 |
98
|
iuneq2i |
⊢ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
100 |
99
|
a1i |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
101 |
8 100
|
sseqtrd |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
102 |
101
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
103 |
|
id |
⊢ ( 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) → 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
104 |
|
eqidd |
⊢ ( 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) → { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑥 〉 } ) |
105 |
|
opeq2 |
⊢ ( 𝑦 = 𝑥 → 〈 𝑍 , 𝑦 〉 = 〈 𝑍 , 𝑥 〉 ) |
106 |
105
|
sneqd |
⊢ ( 𝑦 = 𝑥 → { 〈 𝑍 , 𝑦 〉 } = { 〈 𝑍 , 𝑥 〉 } ) |
107 |
106
|
rspceeqv |
⊢ ( ( 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑥 〉 } ) → ∃ 𝑦 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) |
108 |
103 104 107
|
syl2anc |
⊢ ( 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) → ∃ 𝑦 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) |
109 |
108
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → ∃ 𝑦 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) |
110 |
|
elixpsn |
⊢ ( 𝑍 ∈ 𝑉 → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ↔ ∃ 𝑦 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) ) |
111 |
2 110
|
syl |
⊢ ( 𝜑 → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ↔ ∃ 𝑦 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) ) |
112 |
111
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ↔ ∃ 𝑦 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) ) |
113 |
109 112
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
114 |
3
|
eqcomi |
⊢ { 𝑍 } = 𝑋 |
115 |
|
ixpeq1 |
⊢ ( { 𝑍 } = 𝑋 → X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
116 |
114 115
|
ax-mp |
⊢ X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) |
117 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑍 ) ) |
118 |
93 117
|
syl |
⊢ ( 𝑘 ∈ 𝑋 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑍 ) ) |
119 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑍 ) ) |
120 |
93 119
|
syl |
⊢ ( 𝑘 ∈ 𝑋 → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑍 ) ) |
121 |
118 120
|
oveq12d |
⊢ ( 𝑘 ∈ 𝑋 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
122 |
121
|
eqcomd |
⊢ ( 𝑘 ∈ 𝑋 → ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
123 |
122
|
rgen |
⊢ ∀ 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) |
124 |
|
ixpeq2 |
⊢ ( ∀ 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
125 |
123 124
|
ax-mp |
⊢ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) |
126 |
116 125
|
eqtri |
⊢ X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) |
127 |
126
|
a1i |
⊢ ( 𝜑 → X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
128 |
127
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → X 𝑘 ∈ { 𝑍 } ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
129 |
113 128
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
130 |
102 129
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → { 〈 𝑍 , 𝑥 〉 } ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
131 |
|
eliun |
⊢ ( { 〈 𝑍 , 𝑥 〉 } ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ ∃ 𝑗 ∈ ℕ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
132 |
130 131
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → ∃ 𝑗 ∈ ℕ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
133 |
|
ixpeq1 |
⊢ ( 𝑋 = { 𝑍 } → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
134 |
3 133
|
ax-mp |
⊢ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
135 |
134
|
eleq2i |
⊢ ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
136 |
135
|
biimpi |
⊢ ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
137 |
136
|
adantl |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
138 |
|
elixpsn |
⊢ ( 𝑍 ∈ 𝑉 → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ ∃ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) ) |
139 |
2 138
|
syl |
⊢ ( 𝜑 → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ ∃ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) ) |
140 |
139
|
adantr |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ { 𝑍 } ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ ∃ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) ) |
141 |
137 140
|
mpbid |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ∃ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) |
142 |
|
opex |
⊢ 〈 𝑍 , 𝑥 〉 ∈ V |
143 |
142
|
sneqr |
⊢ ( { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } → 〈 𝑍 , 𝑥 〉 = 〈 𝑍 , 𝑦 〉 ) |
144 |
143
|
adantl |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) → 〈 𝑍 , 𝑥 〉 = 〈 𝑍 , 𝑦 〉 ) |
145 |
|
vex |
⊢ 𝑥 ∈ V |
146 |
145
|
a1i |
⊢ ( 𝜑 → 𝑥 ∈ V ) |
147 |
|
opthg |
⊢ ( ( 𝑍 ∈ 𝑉 ∧ 𝑥 ∈ V ) → ( 〈 𝑍 , 𝑥 〉 = 〈 𝑍 , 𝑦 〉 ↔ ( 𝑍 = 𝑍 ∧ 𝑥 = 𝑦 ) ) ) |
148 |
2 146 147
|
syl2anc |
⊢ ( 𝜑 → ( 〈 𝑍 , 𝑥 〉 = 〈 𝑍 , 𝑦 〉 ↔ ( 𝑍 = 𝑍 ∧ 𝑥 = 𝑦 ) ) ) |
149 |
148
|
adantr |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) → ( 〈 𝑍 , 𝑥 〉 = 〈 𝑍 , 𝑦 〉 ↔ ( 𝑍 = 𝑍 ∧ 𝑥 = 𝑦 ) ) ) |
150 |
144 149
|
mpbid |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) → ( 𝑍 = 𝑍 ∧ 𝑥 = 𝑦 ) ) |
151 |
150
|
simprd |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) → 𝑥 = 𝑦 ) |
152 |
151
|
3adant2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) → 𝑥 = 𝑦 ) |
153 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) → 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
154 |
152 153
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ∧ { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } ) → 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
155 |
154
|
3exp |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → ( { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } → 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) |
156 |
155
|
adantr |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → ( { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } → 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) |
157 |
156
|
rexlimdv |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ∃ 𝑦 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) { 〈 𝑍 , 𝑥 〉 } = { 〈 𝑍 , 𝑦 〉 } → 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
158 |
141 157
|
mpd |
⊢ ( ( 𝜑 ∧ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
159 |
158
|
ex |
⊢ ( 𝜑 → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
160 |
159
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ∧ 𝑗 ∈ ℕ ) → ( { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
161 |
160
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → ( ∃ 𝑗 ∈ ℕ { 〈 𝑍 , 𝑥 〉 } ∈ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → ∃ 𝑗 ∈ ℕ 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
162 |
132 161
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → ∃ 𝑗 ∈ ℕ 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
163 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ ∃ 𝑗 ∈ ℕ 𝑥 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
164 |
162 163
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
165 |
164
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) 𝑥 ∈ ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
166 |
|
dfss3 |
⊢ ( ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ⊆ ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ ∀ 𝑥 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) 𝑥 ∈ ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
167 |
165 166
|
sylibr |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ⊆ ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
168 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
169 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐶 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑖 ) ) |
170 |
169
|
fveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ) |
171 |
170
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) ∧ 𝑗 = 𝑖 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ) |
172 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 𝑖 ∈ ℕ ) |
173 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ V ) |
174 |
168 171 172 173
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ) |
175 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
176 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑖 ) ) |
177 |
176
|
fveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
178 |
177
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) ∧ 𝑗 = 𝑖 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
179 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ V ) |
180 |
175 178 172 179
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
181 |
174 180
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
182 |
181
|
iuneq2dv |
⊢ ( 𝜑 → ∪ 𝑖 ∈ ℕ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) = ∪ 𝑖 ∈ ℕ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
183 |
170 177
|
oveq12d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
184 |
183
|
cbviunv |
⊢ ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ∪ 𝑖 ∈ ℕ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
185 |
184
|
eqcomi |
⊢ ∪ 𝑖 ∈ ℕ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) = ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
186 |
185
|
a1i |
⊢ ( 𝜑 → ∪ 𝑖 ∈ ℕ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) = ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
187 |
182 186
|
eqtr2d |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ℕ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ∪ 𝑖 ∈ ℕ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) |
188 |
167 187
|
sseqtrd |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ⊆ ∪ 𝑖 ∈ ℕ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) |
189 |
188
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ⊆ ∪ 𝑖 ∈ ℕ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) |
190 |
|
fvex |
⊢ ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ V |
191 |
170 84 190
|
fvmpt |
⊢ ( 𝑖 ∈ ℕ → ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ) |
192 |
|
fvex |
⊢ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ V |
193 |
177 87 192
|
fvmpt |
⊢ ( 𝑖 ∈ ℕ → ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
194 |
191 193
|
oveq12d |
⊢ ( 𝑖 ∈ ℕ → ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
195 |
194
|
fveq2d |
⊢ ( 𝑖 ∈ ℕ → ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
196 |
195
|
mpteq2ia |
⊢ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) ) = ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
197 |
|
eqcom |
⊢ ( 𝑗 = 𝑖 ↔ 𝑖 = 𝑗 ) |
198 |
197
|
imbi1i |
⊢ ( ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ↔ ( 𝑖 = 𝑗 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
199 |
|
eqcom |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ↔ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
200 |
199
|
imbi2i |
⊢ ( ( 𝑖 = 𝑗 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ↔ ( 𝑖 = 𝑗 → ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
201 |
198 200
|
bitri |
⊢ ( ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ↔ ( 𝑖 = 𝑗 → ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
202 |
183 201
|
mpbi |
⊢ ( 𝑖 = 𝑗 → ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
203 |
202
|
fveq2d |
⊢ ( 𝑖 = 𝑗 → ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
204 |
203
|
cbvmptv |
⊢ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) = ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
205 |
196 204
|
eqtri |
⊢ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) ) = ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
206 |
205
|
fveq2i |
⊢ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) |
207 |
206
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
208 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) |
209 |
207 208
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) ) ) ∈ ℝ ) |
210 |
|
oveq1 |
⊢ ( 𝑤 = 𝑧 → ( 𝑤 − ( 𝐴 ‘ 𝑍 ) ) = ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) |
211 |
193
|
breq1d |
⊢ ( 𝑖 ∈ ℕ → ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 ↔ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 ) ) |
212 |
211 193
|
ifbieq1d |
⊢ ( 𝑖 ∈ ℕ → if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) = if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) ) |
213 |
191 212
|
oveq12d |
⊢ ( 𝑖 ∈ ℕ → ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) ) ) |
214 |
213
|
fveq2d |
⊢ ( 𝑖 ∈ ℕ → ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) ) ) ) |
215 |
214
|
mpteq2ia |
⊢ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) ) = ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) ) ) ) |
216 |
|
fveq2 |
⊢ ( 𝑖 = ℎ → ( 𝐶 ‘ 𝑖 ) = ( 𝐶 ‘ ℎ ) ) |
217 |
216
|
fveq1d |
⊢ ( 𝑖 = ℎ → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) ) |
218 |
|
fveq2 |
⊢ ( 𝑖 = ℎ → ( 𝐷 ‘ 𝑖 ) = ( 𝐷 ‘ ℎ ) ) |
219 |
218
|
fveq1d |
⊢ ( 𝑖 = ℎ → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) |
220 |
219
|
breq1d |
⊢ ( 𝑖 = ℎ → ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 ↔ ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 ) ) |
221 |
220 219
|
ifbieq1d |
⊢ ( 𝑖 = ℎ → if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) = if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) |
222 |
217 221
|
oveq12d |
⊢ ( 𝑖 = ℎ → ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) ) = ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) ) |
223 |
222
|
fveq2d |
⊢ ( 𝑖 = ℎ → ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) ) ) |
224 |
223
|
cbvmptv |
⊢ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) , 𝑧 ) ) ) ) = ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) ) ) |
225 |
215 224
|
eqtri |
⊢ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) ) = ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) ) ) |
226 |
225
|
a1i |
⊢ ( 𝑤 = 𝑧 → ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) ) = ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) ) ) ) |
227 |
|
breq2 |
⊢ ( 𝑤 = 𝑧 → ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 ↔ ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 ) ) |
228 |
|
id |
⊢ ( 𝑤 = 𝑧 → 𝑤 = 𝑧 ) |
229 |
227 228
|
ifbieq2d |
⊢ ( 𝑤 = 𝑧 → if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) = if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) |
230 |
229
|
eqcomd |
⊢ ( 𝑤 = 𝑧 → if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) = if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) |
231 |
230
|
oveq2d |
⊢ ( 𝑤 = 𝑧 → ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) = ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) |
232 |
231
|
fveq2d |
⊢ ( 𝑤 = 𝑧 → ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) |
233 |
232
|
mpteq2dv |
⊢ ( 𝑤 = 𝑧 → ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑧 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑧 ) ) ) ) = ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) ) |
234 |
226 233
|
eqtr2d |
⊢ ( 𝑤 = 𝑧 → ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) = ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) ) ) |
235 |
234
|
fveq2d |
⊢ ( 𝑤 = 𝑧 → ( Σ^ ‘ ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) ) = ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) ) ) ) |
236 |
210 235
|
breq12d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑤 − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) ) ↔ ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) ) ) ) ) |
237 |
236
|
cbvrabv |
⊢ { 𝑤 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝑤 − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) ) } = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) if ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ≤ 𝑧 , ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) , 𝑧 ) ) ) ) ) } |
238 |
|
eqid |
⊢ sup ( { 𝑤 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝑤 − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) ) } , ℝ , < ) = sup ( { 𝑤 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝑤 − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( ℎ ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ≤ 𝑤 , ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) , 𝑤 ) ) ) ) ) } , ℝ , < ) |
239 |
81 82 83 86 89 189 209 237 238
|
hoidmv1lelem3 |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) [,) ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ‘ 𝑖 ) ) ) ) ) ) |
240 |
239 207
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ∈ ℝ ) → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
241 |
25 80 240
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) = +∞ ) → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
242 |
24 241
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
243 |
1 32 34 4 5
|
hoidmvn0val |
⊢ ( 𝜑 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
244 |
29
|
prodeq1d |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
245 |
|
volicore |
⊢ ( ( ( 𝐴 ‘ 𝑍 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑍 ) ∈ ℝ ) → ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ∈ ℝ ) |
246 |
13 12 245
|
syl2anc |
⊢ ( 𝜑 → ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ∈ ℝ ) |
247 |
246
|
recnd |
⊢ ( 𝜑 → ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ∈ ℂ ) |
248 |
117 119
|
oveq12d |
⊢ ( 𝑘 = 𝑍 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
249 |
248
|
fveq2d |
⊢ ( 𝑘 = 𝑍 → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ) |
250 |
249
|
prodsn |
⊢ ( ( 𝑍 ∈ 𝑉 ∧ ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ∈ ℂ ) → ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ) |
251 |
2 247 250
|
syl2anc |
⊢ ( 𝜑 → ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ) |
252 |
243 244 251
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ) |
253 |
252
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ) |
254 |
|
volico |
⊢ ( ( ( 𝐴 ‘ 𝑍 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑍 ) ∈ ℝ ) → ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) = if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) ) |
255 |
13 12 254
|
syl2anc |
⊢ ( 𝜑 → ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) = if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) ) |
256 |
255
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) = if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) ) |
257 |
|
iftrue |
⊢ ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) → if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) = ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ) |
258 |
257
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) = ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ) |
259 |
253 256 258
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ) |
260 |
59
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
261 |
260
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) |
262 |
259 261
|
breq12d |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) ) ) ) |
263 |
242 262
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
264 |
243
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
265 |
244
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
266 |
251
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ) |
267 |
255
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( vol ‘ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) = if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) ) |
268 |
|
iffalse |
⊢ ( ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) → if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) = 0 ) |
269 |
268
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → if ( ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) , ( ( 𝐵 ‘ 𝑍 ) − ( 𝐴 ‘ 𝑍 ) ) , 0 ) = 0 ) |
270 |
266 267 269
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ∏ 𝑘 ∈ { 𝑍 } ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = 0 ) |
271 |
264 265 270
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) = 0 ) |
272 |
27
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
273 |
272 76
|
sge0ge0 |
⊢ ( 𝜑 → 0 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
274 |
273
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → 0 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
275 |
271 274
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐴 ‘ 𝑍 ) < ( 𝐵 ‘ 𝑍 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
276 |
263 275
|
pm2.61dan |
⊢ ( 𝜑 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |