Step |
Hyp |
Ref |
Expression |
1 |
|
ovnlecvr2.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
ovnlecvr2.c |
⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
3 |
|
ovnlecvr2.d |
⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
4 |
|
ovnlecvr2.s |
⊢ ( 𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
5 |
|
ovnlecvr2.l |
⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) |
6 |
|
fveq2 |
⊢ ( 𝑋 = ∅ → ( voln* ‘ 𝑋 ) = ( voln* ‘ ∅ ) ) |
7 |
6
|
fveq1d |
⊢ ( 𝑋 = ∅ → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) = ( ( voln* ‘ ∅ ) ‘ 𝐴 ) ) |
8 |
7
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) = ( ( voln* ‘ ∅ ) ‘ 𝐴 ) ) |
9 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
10 |
|
1nn |
⊢ 1 ∈ ℕ |
11 |
|
ne0i |
⊢ ( 1 ∈ ℕ → ℕ ≠ ∅ ) |
12 |
10 11
|
ax-mp |
⊢ ℕ ≠ ∅ |
13 |
12
|
a1i |
⊢ ( 𝜑 → ℕ ≠ ∅ ) |
14 |
|
iunconst |
⊢ ( ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ { ∅ } = { ∅ } ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ℕ { ∅ } = { ∅ } ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ∪ 𝑗 ∈ ℕ { ∅ } = { ∅ } ) |
17 |
|
ixpeq1 |
⊢ ( 𝑋 = ∅ → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ∅ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
18 |
|
ixp0x |
⊢ X 𝑘 ∈ ∅ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = { ∅ } |
19 |
18
|
a1i |
⊢ ( 𝑋 = ∅ → X 𝑘 ∈ ∅ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = { ∅ } ) |
20 |
17 19
|
eqtrd |
⊢ ( 𝑋 = ∅ → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = { ∅ } ) |
21 |
20
|
adantr |
⊢ ( ( 𝑋 = ∅ ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = { ∅ } ) |
22 |
21
|
iuneq2dv |
⊢ ( 𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ { ∅ } ) |
23 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ { ∅ } ) |
24 |
|
reex |
⊢ ℝ ∈ V |
25 |
|
mapdm0 |
⊢ ( ℝ ∈ V → ( ℝ ↑m ∅ ) = { ∅ } ) |
26 |
24 25
|
ax-mp |
⊢ ( ℝ ↑m ∅ ) = { ∅ } |
27 |
26
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ℝ ↑m ∅ ) = { ∅ } ) |
28 |
16 23 27
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ℝ ↑m ∅ ) ) |
29 |
9 28
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → 𝐴 ⊆ ( ℝ ↑m ∅ ) ) |
30 |
29
|
ovn0val |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ ∅ ) ‘ 𝐴 ) = 0 ) |
31 |
8 30
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) = 0 ) |
32 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
33 |
|
nnex |
⊢ ℕ ∈ V |
34 |
33
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
35 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
36 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑋 ∈ Fin ) |
37 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) ) |
38 |
|
elmapi |
⊢ ( ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) → ( 𝐶 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
39 |
37 38
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
40 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) ) |
41 |
|
elmapi |
⊢ ( ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑋 ) → ( 𝐷 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
42 |
40 41
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
43 |
5 36 39 42
|
hoidmvcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) ) |
44 |
35 43
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ∈ ( 0 [,] +∞ ) ) |
45 |
32 34 44
|
sge0ge0mpt |
⊢ ( 𝜑 → 0 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
46 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → 0 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
47 |
31 46
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
48 |
|
simpl |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → 𝜑 ) |
49 |
|
neqne |
⊢ ( ¬ 𝑋 = ∅ → 𝑋 ≠ ∅ ) |
50 |
49
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → 𝑋 ≠ ∅ ) |
51 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ∈ Fin ) |
52 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝑋 ≠ ∅ ) |
53 |
39
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ) |
54 |
42
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ) |
55 |
54
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ* ) |
56 |
|
icossre |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ* ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ℝ ) |
57 |
53 55 56
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ℝ ) |
58 |
57
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∀ 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ℝ ) |
59 |
|
ss2ixp |
⊢ ( ∀ 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ℝ → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ X 𝑘 ∈ 𝑋 ℝ ) |
60 |
58 59
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ X 𝑘 ∈ 𝑋 ℝ ) |
61 |
24
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
62 |
|
ixpconstg |
⊢ ( ( 𝑋 ∈ Fin ∧ ℝ ∈ V ) → X 𝑘 ∈ 𝑋 ℝ = ( ℝ ↑m 𝑋 ) ) |
63 |
1 61 62
|
syl2anc |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ℝ = ( ℝ ↑m 𝑋 ) ) |
64 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ℝ = ( ℝ ↑m 𝑋 ) ) |
65 |
60 64
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ( ℝ ↑m 𝑋 ) ) |
66 |
65
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ( ℝ ↑m 𝑋 ) ) |
67 |
|
iunss |
⊢ ( ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ( ℝ ↑m 𝑋 ) ↔ ∀ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ( ℝ ↑m 𝑋 ) ) |
68 |
66 67
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ⊆ ( ℝ ↑m 𝑋 ) ) |
69 |
4 68
|
sstrd |
⊢ ( 𝜑 → 𝐴 ⊆ ( ℝ ↑m 𝑋 ) ) |
70 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝐴 ⊆ ( ℝ ↑m 𝑋 ) ) |
71 |
|
eqid |
⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } |
72 |
51 52 70 71
|
ovnn0val |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) = inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) |
73 |
|
ssrab2 |
⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ⊆ ℝ* |
74 |
73
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ⊆ ℝ* ) |
75 |
32 34 44
|
sge0xrclmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ* ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ* ) |
77 |
|
opelxpi |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ) → 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ∈ ( ℝ × ℝ ) ) |
78 |
53 54 77
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ∈ ( ℝ × ℝ ) ) |
79 |
78
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
80 |
24 24
|
xpex |
⊢ ( ℝ × ℝ ) ∈ V |
81 |
80
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ℝ × ℝ ) ∈ V ) |
82 |
|
elmapg |
⊢ ( ( ( ℝ × ℝ ) ∈ V ∧ 𝑋 ∈ Fin ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
83 |
81 36 82
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
84 |
79 83
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
85 |
84
|
fmpttd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
86 |
|
ovexd |
⊢ ( 𝜑 → ( ( ℝ × ℝ ) ↑m 𝑋 ) ∈ V ) |
87 |
|
elmapg |
⊢ ( ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ∈ V ∧ ℕ ∈ V ) → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ↔ ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
88 |
86 34 87
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ↔ ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
89 |
85 88
|
mpbird |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
90 |
89
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
91 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ ) |
92 |
|
mptexg |
⊢ ( 𝑋 ∈ Fin → ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ∈ V ) |
93 |
1 92
|
syl |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ∈ V ) |
94 |
93
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ∈ V ) |
95 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
96 |
95
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ∈ V ) → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
97 |
91 94 96
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
98 |
97
|
coeq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) = ( [,) ∘ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ) |
99 |
98
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑘 ) ) |
100 |
99
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑘 ) ) |
101 |
79
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
102 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 ) |
103 |
101 102
|
fvovco |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑘 ) = ( ( 1st ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) ) ) |
104 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 ) |
105 |
|
opex |
⊢ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ∈ V |
106 |
105
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ∈ V ) |
107 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) = ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) |
108 |
107
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝑋 ∧ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ∈ V ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) = 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) |
109 |
104 106 108
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) = 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) |
110 |
109
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) = ( 1st ‘ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
111 |
|
fvex |
⊢ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ∈ V |
112 |
|
fvex |
⊢ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ∈ V |
113 |
|
op1stg |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ∈ V ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ∈ V ) → ( 1st ‘ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
114 |
111 112 113
|
mp2an |
⊢ ( 1st ‘ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) |
115 |
114
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
116 |
110 115
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
117 |
109
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) = ( 2nd ‘ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
118 |
111 112
|
op2nd |
⊢ ( 2nd ‘ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) |
119 |
118
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
120 |
117 119
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
121 |
116 120
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
122 |
121
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ‘ 𝑘 ) ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
123 |
100 103 122
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
124 |
123
|
ixpeq2dva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
125 |
124
|
iuneq2dv |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
126 |
4 125
|
sseqtrd |
⊢ ( 𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
127 |
126
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
128 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) ) |
129 |
51
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → 𝑋 ∈ Fin ) |
130 |
52
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → 𝑋 ≠ ∅ ) |
131 |
39
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
132 |
42
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
133 |
5 129 130 131 132
|
hoidmvn0val |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
134 |
133
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) |
135 |
134
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) ) |
136 |
123
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
137 |
136
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
138 |
137
|
prodeq2dv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
139 |
138
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) |
140 |
139
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) ) |
141 |
140
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) ) |
142 |
128 135 141
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
143 |
127 142
|
jca |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
144 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑖 |
145 |
|
nfmpt1 |
⊢ Ⅎ 𝑗 ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
146 |
144 145
|
nfeq |
⊢ Ⅎ 𝑗 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
147 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑖 |
148 |
|
nfcv |
⊢ Ⅎ 𝑘 ℕ |
149 |
|
nfmpt1 |
⊢ Ⅎ 𝑘 ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) |
150 |
148 149
|
nfmpt |
⊢ Ⅎ 𝑘 ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
151 |
147 150
|
nfeq |
⊢ Ⅎ 𝑘 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) |
152 |
|
fveq1 |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( 𝑖 ‘ 𝑗 ) = ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) |
153 |
152
|
coeq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) = ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ) |
154 |
153
|
fveq1d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
155 |
154
|
adantr |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
156 |
151 155
|
ixpeq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
157 |
156
|
adantr |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
158 |
146 157
|
iuneq2df |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
159 |
158
|
sseq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
160 |
|
nfv |
⊢ Ⅎ 𝑘 𝑗 ∈ ℕ |
161 |
151 160
|
nfan |
⊢ Ⅎ 𝑘 ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∧ 𝑗 ∈ ℕ ) |
162 |
154
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
163 |
162
|
a1d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( 𝑘 ∈ 𝑋 → ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
164 |
163
|
adantr |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 → ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
165 |
161 164
|
ralrimi |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∧ 𝑗 ∈ ℕ ) → ∀ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
166 |
165
|
prodeq2d |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
167 |
146 166
|
mpteq2da |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
168 |
167
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
169 |
168
|
eqeq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
170 |
159 169
|
anbi12d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) → ( ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ↔ ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
171 |
170
|
rspcev |
⊢ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) → ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
172 |
90 143 171
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
173 |
76 172
|
jca |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ* ∧ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
174 |
|
eqeq1 |
⊢ ( 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) → ( 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
175 |
174
|
anbi2d |
⊢ ( 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) → ( ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ↔ ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
176 |
175
|
rexbidv |
⊢ ( 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) → ( ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ↔ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
177 |
176
|
elrab |
⊢ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ↔ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ* ∧ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
178 |
173 177
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ) |
179 |
|
infxrlb |
⊢ ( ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ⊆ ℝ* ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ) → inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
180 |
74 178 179
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
181 |
72 180
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
182 |
48 50 181
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
183 |
47 182
|
pm2.61dan |
⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) |