Proof of Theorem ovnovollem2
Step | Hyp | Ref
| Expression |
1 | | ovnovollem2.i |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ)
↑𝑚 {𝐴}) ↑𝑚
ℕ)) |
2 | | elmapi 7879 |
. . . . . . . . 9
⊢ (𝐼 ∈ (((ℝ ×
ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)
→ 𝐼:ℕ⟶((ℝ × ℝ)
↑𝑚 {𝐴})) |
3 | 1, 2 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ)
↑𝑚 {𝐴})) |
4 | 3 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ)
↑𝑚 {𝐴})) |
5 | | simpr 477 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
6 | 4, 5 | ffvelrnd 6360 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ)
↑𝑚 {𝐴})) |
7 | | elmapi 7879 |
. . . . . 6
⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ)
↑𝑚 {𝐴}) → (𝐼‘𝑗):{𝐴}⟶(ℝ ×
ℝ)) |
8 | 6, 7 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):{𝐴}⟶(ℝ ×
ℝ)) |
9 | | ovnovollem2.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
10 | | snidg 4206 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) |
11 | 9, 10 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
12 | 11 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴}) |
13 | 8, 12 | ffvelrnd 6360 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ (ℝ ×
ℝ)) |
14 | | ovnovollem2.f |
. . . 4
⊢ 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼‘𝑗)‘𝐴)) |
15 | 13, 14 | fmptd 6385 |
. . 3
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ×
ℝ)) |
16 | | reex 10027 |
. . . . . 6
⊢ ℝ
∈ V |
17 | 16, 16 | xpex 6962 |
. . . . 5
⊢ (ℝ
× ℝ) ∈ V |
18 | | nnex 11026 |
. . . . 5
⊢ ℕ
∈ V |
19 | 17, 18 | elmap 7886 |
. . . 4
⊢ (𝐹 ∈ ((ℝ ×
ℝ) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶(ℝ ×
ℝ)) |
20 | 19 | a1i 11 |
. . 3
⊢ (𝜑 → (𝐹 ∈ ((ℝ × ℝ)
↑𝑚 ℕ) ↔ 𝐹:ℕ⟶(ℝ ×
ℝ))) |
21 | 15, 20 | mpbird 247 |
. 2
⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ)
↑𝑚 ℕ)) |
22 | | ovnovollem2.s |
. . . . . 6
⊢ (𝜑 → (𝐵 ↑𝑚 {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
23 | | elsni 4194 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴) |
24 | 23 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝐴)) |
25 | 24 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝐴)) |
26 | | elmapfun 7881 |
. . . . . . . . . . . . . 14
⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ)
↑𝑚 {𝐴}) → Fun (𝐼‘𝑗)) |
27 | 6, 26 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Fun (𝐼‘𝑗)) |
28 | | fdm 6051 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼‘𝑗):{𝐴}⟶(ℝ × ℝ) →
dom (𝐼‘𝑗) = {𝐴}) |
29 | 8, 28 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → dom (𝐼‘𝑗) = {𝐴}) |
30 | 29 | eqcomd 2628 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → {𝐴} = dom (𝐼‘𝑗)) |
31 | 12, 30 | eleqtrd 2703 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼‘𝑗)) |
32 | | fvco 6274 |
. . . . . . . . . . . . 13
⊢ ((Fun
(𝐼‘𝑗) ∧ 𝐴 ∈ dom (𝐼‘𝑗)) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴))) |
33 | 27, 31, 32 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴))) |
34 | 33 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴))) |
35 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℕ) |
36 | | fvexd 6203 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) ∈ V) |
37 | 14 | fvmpt2 6291 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑗 ∈ ℕ ∧ ((𝐼‘𝑗)‘𝐴) ∈ V) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴)) |
38 | 35, 36, 37 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴)) |
39 | 38 | eqcomd 2628 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) = (𝐹‘𝑗)) |
40 | 39 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ →
([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗))) |
41 | 40 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗))) |
42 | 15 | ffund 6049 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → Fun 𝐹) |
43 | 42 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Fun 𝐹) |
44 | 14, 13 | dmmptd 6024 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → dom 𝐹 = ℕ) |
45 | 44 | eqcomd 2628 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ℕ = dom 𝐹) |
46 | 45 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ℕ = dom 𝐹) |
47 | 5, 46 | eleqtrd 2703 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹) |
48 | | fvco 6274 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐹 ∧ 𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗))) |
49 | 43, 47, 48 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗))) |
50 | 49 | eqcomd 2628 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = (([,) ∘ 𝐹)‘𝑗)) |
51 | 41, 50 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘((𝐼‘𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗)) |
52 | 51 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼‘𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗)) |
53 | 25, 34, 52 | 3eqtrd 2660 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗)) |
54 | 53 | ixpeq2dva 7923 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
{𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗)) |
55 | | snex 4908 |
. . . . . . . . . . 11
⊢ {𝐴} ∈ V |
56 | | fvex 6201 |
. . . . . . . . . . 11
⊢ (([,)
∘ 𝐹)‘𝑗) ∈ V |
57 | 55, 56 | ixpconst 7918 |
. . . . . . . . . 10
⊢ X𝑘 ∈
{𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}) |
58 | 57 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
{𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})) |
59 | 54, 58 | eqtrd 2656 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
{𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})) |
60 | 59 | iuneq2dv 4542 |
. . . . . . 7
⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ ((([,) ∘
𝐹)‘𝑗) ↑𝑚 {𝐴})) |
61 | | nfv 1843 |
. . . . . . . 8
⊢
Ⅎ𝑗𝜑 |
62 | 18 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℕ ∈
V) |
63 | | fvexd 6203 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V) |
64 | 61, 62, 63, 9 | iunmapsn 39409 |
. . . . . . 7
⊢ (𝜑 → ∪ 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}) = (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})) |
65 | 60, 64 | eqtrd 2656 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = (∪
𝑗 ∈ ℕ (([,)
∘ 𝐹)‘𝑗) ↑𝑚
{𝐴})) |
66 | 22, 65 | sseqtrd 3641 |
. . . . 5
⊢ (𝜑 → (𝐵 ↑𝑚 {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})) |
67 | | ovnovollem2.b |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
68 | 18, 56 | iunex 7147 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V |
69 | 68 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V) |
70 | 55 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {𝐴} ∈ V) |
71 | | ne0i 3921 |
. . . . . . 7
⊢ (𝐴 ∈ {𝐴} → {𝐴} ≠ ∅) |
72 | 11, 71 | syl 17 |
. . . . . 6
⊢ (𝜑 → {𝐴} ≠ ∅) |
73 | 67, 69, 70, 72 | mapss2 39397 |
. . . . 5
⊢ (𝜑 → (𝐵 ⊆ ∪
𝑗 ∈ ℕ (([,)
∘ 𝐹)‘𝑗) ↔ (𝐵 ↑𝑚 {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))) |
74 | 66, 73 | mpbird 247 |
. . . 4
⊢ (𝜑 → 𝐵 ⊆ ∪
𝑗 ∈ ℕ (([,)
∘ 𝐹)‘𝑗)) |
75 | | icof 39411 |
. . . . . . . 8
⊢
[,):(ℝ* × ℝ*)⟶𝒫
ℝ* |
76 | 75 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → [,):(ℝ*
× ℝ*)⟶𝒫
ℝ*) |
77 | | rexpssxrxp 10084 |
. . . . . . . 8
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
78 | 77 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (ℝ × ℝ)
⊆ (ℝ* × ℝ*)) |
79 | 76, 78, 15 | fcoss 39402 |
. . . . . 6
⊢ (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫
ℝ*) |
80 | 79 | ffnd 6046 |
. . . . 5
⊢ (𝜑 → ([,) ∘ 𝐹) Fn ℕ) |
81 | | fniunfv 6505 |
. . . . 5
⊢ (([,)
∘ 𝐹) Fn ℕ
→ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ∪ ran ([,)
∘ 𝐹)) |
82 | 80, 81 | syl 17 |
. . . 4
⊢ (𝜑 → ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ∪ ran ([,)
∘ 𝐹)) |
83 | 74, 82 | sseqtrd 3641 |
. . 3
⊢ (𝜑 → 𝐵 ⊆ ∪ ran
([,) ∘ 𝐹)) |
84 | | ovnovollem2.z |
. . . 4
⊢ (𝜑 → 𝑍 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))) |
85 | | nfcv 2764 |
. . . . . . 7
⊢
Ⅎ𝑗𝐹 |
86 | | ressxr 10083 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℝ* |
87 | | xpss2 5229 |
. . . . . . . . . 10
⊢ (ℝ
⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ
× ℝ*)) |
88 | 86, 87 | ax-mp 5 |
. . . . . . . . 9
⊢ (ℝ
× ℝ) ⊆ (ℝ × ℝ*) |
89 | 88 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (ℝ × ℝ)
⊆ (ℝ × ℝ*)) |
90 | 15, 89 | fssd 6057 |
. . . . . . 7
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ×
ℝ*)) |
91 | 85, 90 | volicofmpt 40214 |
. . . . . 6
⊢ (𝜑 → ((vol ∘ [,)) ∘
𝐹) = (𝑗 ∈ ℕ ↦
(vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))))) |
92 | 9 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ 𝑉) |
93 | | fvexd 6203 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ V) |
94 | 5, 93, 37 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴)) |
95 | 94, 13 | eqeltrd 2701 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) ∈ (ℝ ×
ℝ)) |
96 | | 1st2nd2 7205 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) →
(𝐹‘𝑗) = 〈(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))〉) |
97 | 95, 96 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = 〈(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))〉) |
98 | 97 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = ([,)‘〈(1st
‘(𝐹‘𝑗)), (2nd
‘(𝐹‘𝑗))〉)) |
99 | | df-ov 6653 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))) = ([,)‘〈(1st
‘(𝐹‘𝑗)), (2nd
‘(𝐹‘𝑗))〉) |
100 | 99 | eqcomi 2631 |
. . . . . . . . . . . . . . 15
⊢
([,)‘〈(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))〉) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))) |
101 | 100 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) →
([,)‘〈(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))〉) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) |
102 | 49, 98, 101 | 3eqtrd 2660 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) |
103 | 33, 51, 102 | 3eqtrd 2660 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) |
104 | 103 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,)
∘ (𝐼‘𝑗))‘𝐴)) = (vol‘((1st
‘(𝐹‘𝑗))[,)(2nd
‘(𝐹‘𝑗))))) |
105 | | xp1st 7198 |
. . . . . . . . . . . . 13
⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) →
(1st ‘(𝐹‘𝑗)) ∈ ℝ) |
106 | 95, 105 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (1st
‘(𝐹‘𝑗)) ∈
ℝ) |
107 | | xp2nd 7199 |
. . . . . . . . . . . . 13
⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) →
(2nd ‘(𝐹‘𝑗)) ∈ ℝ) |
108 | 95, 107 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2nd
‘(𝐹‘𝑗)) ∈
ℝ) |
109 | | volicore 40795 |
. . . . . . . . . . . 12
⊢
(((1st ‘(𝐹‘𝑗)) ∈ ℝ ∧ (2nd
‘(𝐹‘𝑗)) ∈ ℝ) →
(vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) ∈ ℝ) |
110 | 106, 108,
109 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) →
(vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) ∈ ℝ) |
111 | 104, 110 | eqeltrd 2701 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,)
∘ (𝐼‘𝑗))‘𝐴)) ∈ ℝ) |
112 | 111 | recnd 10068 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,)
∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ) |
113 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐴 → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝐴)) |
114 | 113 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴))) |
115 | 114 | prodsn 14692 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴))) |
116 | 92, 112, 115 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴))) |
117 | 116, 104 | eqtr2d 2657 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) →
(vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) |
118 | 117 | mpteq2dva 4744 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ ℕ ↦
(vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) |
119 | 91, 118 | eqtrd 2656 |
. . . . 5
⊢ (𝜑 → ((vol ∘ [,)) ∘
𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) |
120 | 119 | fveq2d 6195 |
. . . 4
⊢ (𝜑 →
(Σ^‘((vol ∘ [,)) ∘ 𝐹)) =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))) |
121 | 84, 120 | eqtr4d 2659 |
. . 3
⊢ (𝜑 → 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝐹))) |
122 | 83, 121 | jca 554 |
. 2
⊢ (𝜑 → (𝐵 ⊆ ∪ ran
([,) ∘ 𝐹) ∧ 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝐹)))) |
123 | | coeq2 5280 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹)) |
124 | 123 | rneqd 5353 |
. . . . . 6
⊢ (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹)) |
125 | 124 | unieqd 4446 |
. . . . 5
⊢ (𝑓 = 𝐹 → ∪ ran
([,) ∘ 𝑓) = ∪ ran ([,) ∘ 𝐹)) |
126 | 125 | sseq2d 3633 |
. . . 4
⊢ (𝑓 = 𝐹 → (𝐵 ⊆ ∪ ran
([,) ∘ 𝑓) ↔
𝐵 ⊆ ∪ ran ([,) ∘ 𝐹))) |
127 | | coeq2 5280 |
. . . . . 6
⊢ (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘
𝐹)) |
128 | 127 | fveq2d 6195 |
. . . . 5
⊢ (𝑓 = 𝐹 →
(Σ^‘((vol ∘ [,)) ∘ 𝑓)) =
(Σ^‘((vol ∘ [,)) ∘ 𝐹))) |
129 | 128 | eqeq2d 2632 |
. . . 4
⊢ (𝑓 = 𝐹 → (𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝐹)))) |
130 | 126, 129 | anbi12d 747 |
. . 3
⊢ (𝑓 = 𝐹 → ((𝐵 ⊆ ∪ ran
([,) ∘ 𝑓) ∧ 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ⊆ ∪ ran
([,) ∘ 𝐹) ∧ 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝐹))))) |
131 | 130 | rspcev 3309 |
. 2
⊢ ((𝐹 ∈ ((ℝ ×
ℝ) ↑𝑚 ℕ) ∧ (𝐵 ⊆ ∪ ran
([,) ∘ 𝐹) ∧ 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝐹)))) → ∃𝑓 ∈ ((ℝ ×
ℝ) ↑𝑚 ℕ)(𝐵 ⊆ ∪ ran
([,) ∘ 𝑓) ∧ 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝑓)))) |
132 | 21, 122, 131 | syl2anc 693 |
1
⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ)
↑𝑚 ℕ)(𝐵 ⊆ ∪ ran
([,) ∘ 𝑓) ∧ 𝑍 =
(Σ^‘((vol ∘ [,)) ∘ 𝑓)))) |