MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg1climres Structured version   Visualization version   GIF version

Theorem itg1climres 23786
Description: Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
itg1climres.1 (𝜑𝐴:ℕ⟶dom vol)
itg1climres.2 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)))
itg1climres.3 (𝜑 ran 𝐴 = ℝ)
itg1climres.4 (𝜑𝐹 ∈ dom ∫1)
itg1climres.5 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
Assertion
Ref Expression
itg1climres (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ⇝ (∫1𝐹))
Distinct variable groups:   𝑥,𝑛,𝐴   𝑛,𝐹,𝑥   𝜑,𝑛,𝑥
Allowed substitution hints:   𝐺(𝑥,𝑛)

Proof of Theorem itg1climres
Dummy variables 𝑗 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 11928 . . 3 ℕ = (ℤ‘1)
2 1zzd 11660 . . 3 (𝜑 → 1 ∈ ℤ)
3 itg1climres.4 . . . . 5 (𝜑𝐹 ∈ dom ∫1)
4 i1frn 23749 . . . . 5 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
53, 4syl 17 . . . 4 (𝜑 → ran 𝐹 ∈ Fin)
6 difss 3901 . . . 4 (ran 𝐹 ∖ {0}) ⊆ ran 𝐹
7 ssfi 8391 . . . 4 ((ran 𝐹 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin)
85, 6, 7sylancl 580 . . 3 (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin)
9 1zzd 11660 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 1 ∈ ℤ)
10 i1fima 23750 . . . . . . . . . . . 12 (𝐹 ∈ dom ∫1 → (𝐹 “ {𝑘}) ∈ dom vol)
113, 10syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ {𝑘}) ∈ dom vol)
1211ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐹 “ {𝑘}) ∈ dom vol)
13 itg1climres.1 . . . . . . . . . . . 12 (𝜑𝐴:ℕ⟶dom vol)
1413ffvelrnda 6553 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ dom vol)
1514adantlr 706 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ dom vol)
16 inmbl 23614 . . . . . . . . . 10 (((𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴𝑛) ∈ dom vol) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ∈ dom vol)
1712, 15, 16syl2anc 579 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ∈ dom vol)
18 mblvol 23602 . . . . . . . . 9 (((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ∈ dom vol → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
1917, 18syl 17 . . . . . . . 8 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
20 inss1 3994 . . . . . . . . . 10 ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘})
2120a1i 11 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘}))
22 mblss 23603 . . . . . . . . . 10 ((𝐹 “ {𝑘}) ∈ dom vol → (𝐹 “ {𝑘}) ⊆ ℝ)
2312, 22syl 17 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐹 “ {𝑘}) ⊆ ℝ)
24 mblvol 23602 . . . . . . . . . . 11 ((𝐹 “ {𝑘}) ∈ dom vol → (vol‘(𝐹 “ {𝑘})) = (vol*‘(𝐹 “ {𝑘})))
2512, 24syl 17 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹 “ {𝑘})) = (vol*‘(𝐹 “ {𝑘})))
26 i1fima2sn 23752 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) ∈ ℝ)
273, 26sylan 575 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) ∈ ℝ)
2827adantr 472 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹 “ {𝑘})) ∈ ℝ)
2925, 28eqeltrrd 2845 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝐹 “ {𝑘})) ∈ ℝ)
30 ovolsscl 23558 . . . . . . . . 9 ((((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘}) ∧ (𝐹 “ {𝑘}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑘})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ∈ ℝ)
3121, 23, 29, 30syl3anc 1490 . . . . . . . 8 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ∈ ℝ)
3219, 31eqeltrd 2844 . . . . . . 7 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ∈ ℝ)
3332fmpttd 6579 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))):ℕ⟶ℝ)
34 itg1climres.2 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)))
3534adantlr 706 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)))
36 sslin 4000 . . . . . . . . . . . 12 ((𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
3735, 36syl 17 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
3813adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴:ℕ⟶dom vol)
39 peano2nn 11292 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40 ffvelrn 6551 . . . . . . . . . . . . . 14 ((𝐴:ℕ⟶dom vol ∧ (𝑛 + 1) ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
4138, 39, 40syl2an 589 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
42 inmbl 23614 . . . . . . . . . . . . 13 (((𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘(𝑛 + 1)) ∈ dom vol) → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
4312, 41, 42syl2anc 579 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
44 mblss 23603 . . . . . . . . . . . 12 (((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ)
4543, 44syl 17 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ)
46 ovolss 23557 . . . . . . . . . . 11 ((((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∧ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
4737, 45, 46syl2anc 579 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
48 mblvol 23602 . . . . . . . . . . 11 (((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
4943, 48syl 17 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
5047, 19, 493brtr4d 4843 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
5150ralrimiva 3113 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
52 fveq2 6379 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
5352ineq2d 3978 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))
5453fveq2d 6383 . . . . . . . . . . . 12 (𝑛 = 𝑗 → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))))
55 eqid 2765 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
56 fvex 6392 . . . . . . . . . . . 12 (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ∈ V
5754, 55, 56fvmpt 6475 . . . . . . . . . . 11 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))))
58 peano2nn 11292 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
59 fveq2 6379 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (𝐴𝑛) = (𝐴‘(𝑗 + 1)))
6059ineq2d 3978 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
6160fveq2d 6383 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
62 fvex 6392 . . . . . . . . . . . . 13 (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))) ∈ V
6361, 55, 62fvmpt 6475 . . . . . . . . . . . 12 ((𝑗 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
6458, 63syl 17 . . . . . . . . . . 11 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
6557, 64breq12d 4824 . . . . . . . . . 10 (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)) ↔ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))))
6665ralbiia 3126 . . . . . . . . 9 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)) ↔ ∀𝑗 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
67 fvoveq1 6869 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑗 + 1)))
6867ineq2d 3978 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) = ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
6968fveq2d 6383 . . . . . . . . . . 11 (𝑛 = 𝑗 → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
7054, 69breq12d 4824 . . . . . . . . . 10 (𝑛 = 𝑗 → ((vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) ↔ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))))
7170cbvralv 3319 . . . . . . . . 9 (∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) ↔ ∀𝑗 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
7266, 71bitr4i 269 . . . . . . . 8 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)) ↔ ∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
7351, 72sylibr 225 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)))
7473r19.21bi 3079 . . . . . 6 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)))
75 ovolss 23557 . . . . . . . . . . 11 ((((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘}) ∧ (𝐹 “ {𝑘}) ⊆ ℝ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol*‘(𝐹 “ {𝑘})))
7620, 23, 75sylancr 581 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol*‘(𝐹 “ {𝑘})))
7776, 19, 253brtr4d 4843 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})))
7877ralrimiva 3113 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})))
7957breq1d 4821 . . . . . . . . . 10 (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})) ↔ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘}))))
8079ralbiia 3126 . . . . . . . . 9 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘})))
8154breq1d 4821 . . . . . . . . . 10 (𝑛 = 𝑗 → ((vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})) ↔ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘}))))
8281cbvralv 3319 . . . . . . . . 9 (∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘})))
8380, 82bitr4i 269 . . . . . . . 8 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})) ↔ ∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})))
8478, 83sylibr 225 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})))
85 brralrspcev 4871 . . . . . . 7 (((vol‘(𝐹 “ {𝑘})) ∈ ℝ ∧ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘}))) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥)
8627, 84, 85syl2anc 579 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥)
871, 9, 33, 74, 86climsup 14699 . . . . 5 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ))
8817fmpttd 6579 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))):ℕ⟶dom vol)
8937ralrimiva 3113 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
90 eqid 2765 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))
91 fvex 6392 . . . . . . . . . . . . 13 (𝐴𝑗) ∈ V
9291inex2 4963 . . . . . . . . . . . 12 ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ∈ V
9353, 90, 92fvmpt 6475 . . . . . . . . . . 11 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))
94 fvex 6392 . . . . . . . . . . . . . 14 (𝐴‘(𝑗 + 1)) ∈ V
9594inex2 4963 . . . . . . . . . . . . 13 ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))) ∈ V
9660, 90, 95fvmpt 6475 . . . . . . . . . . . 12 ((𝑗 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1)) = ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
9758, 96syl 17 . . . . . . . . . . 11 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1)) = ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
9893, 97sseq12d 3796 . . . . . . . . . 10 (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1)) ↔ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
9998ralbiia 3126 . . . . . . . . 9 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1)) ↔ ∀𝑗 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
10053, 68sseq12d 3796 . . . . . . . . . 10 (𝑛 = 𝑗 → (((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
101100cbvralv 3319 . . . . . . . . 9 (∀𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ∀𝑗 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
10299, 101bitr4i 269 . . . . . . . 8 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1)) ↔ ∀𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
10389, 102sylibr 225 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1)))
104 volsup 23628 . . . . . . 7 (((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))):ℕ⟶dom vol ∧ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1))) → (vol‘ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ))
10588, 103, 104syl2anc 579 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ))
10693iuneq2i 4697 . . . . . . . . . 10 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = 𝑗 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗))
10753cbviunv 4717 . . . . . . . . . 10 𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = 𝑗 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗))
108 iunin2 4742 . . . . . . . . . 10 𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛))
109106, 107, 1083eqtr2i 2793 . . . . . . . . 9 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛))
110 ffn 6225 . . . . . . . . . . . . . 14 (𝐴:ℕ⟶dom vol → 𝐴 Fn ℕ)
111 fniunfv 6701 . . . . . . . . . . . . . 14 (𝐴 Fn ℕ → 𝑛 ∈ ℕ (𝐴𝑛) = ran 𝐴)
11213, 110, 1113syl 18 . . . . . . . . . . . . 13 (𝜑 𝑛 ∈ ℕ (𝐴𝑛) = ran 𝐴)
113 itg1climres.3 . . . . . . . . . . . . 13 (𝜑 ran 𝐴 = ℝ)
114112, 113eqtrd 2799 . . . . . . . . . . . 12 (𝜑 𝑛 ∈ ℕ (𝐴𝑛) = ℝ)
115114adantr 472 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑛 ∈ ℕ (𝐴𝑛) = ℝ)
116115ineq2d 3978 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ ℝ))
11711adantr 472 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑘}) ∈ dom vol)
118117, 22syl 17 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑘}) ⊆ ℝ)
119 df-ss 3748 . . . . . . . . . . 11 ((𝐹 “ {𝑘}) ⊆ ℝ ↔ ((𝐹 “ {𝑘}) ∩ ℝ) = (𝐹 “ {𝑘}))
120118, 119sylib 209 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑘}) ∩ ℝ) = (𝐹 “ {𝑘}))
121116, 120eqtrd 2799 . . . . . . . . 9 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛)) = (𝐹 “ {𝑘}))
122109, 121syl5eq 2811 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = (𝐹 “ {𝑘}))
123 ffn 6225 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))):ℕ⟶dom vol → (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) Fn ℕ)
124 fniunfv 6701 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) Fn ℕ → 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
12588, 123, 1243syl 18 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
126122, 125eqtr3d 2801 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑘}) = ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
127126fveq2d 6383 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) = (vol‘ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
12833frnd 6232 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ⊆ ℝ)
12933fdmd 6234 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ℕ)
130 1nn 11291 . . . . . . . . . . 11 1 ∈ ℕ
131 ne0i 4087 . . . . . . . . . . 11 (1 ∈ ℕ → ℕ ≠ ∅)
132130, 131mp1i 13 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ℕ ≠ ∅)
133129, 132eqnetrd 3004 . . . . . . . . 9 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅)
134 dm0rn0 5512 . . . . . . . . . 10 (dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ∅)
135134necon3bii 2989 . . . . . . . . 9 (dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅)
136133, 135sylib 209 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅)
137 ffn 6225 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) Fn ℕ)
138 breq1 4814 . . . . . . . . . . . 12 (𝑧 = ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) → (𝑧𝑥 ↔ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥))
139138ralrn 6556 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))𝑧𝑥 ↔ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥))
14033, 137, 1393syl 18 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))𝑧𝑥 ↔ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥))
141140rexbidv 3199 . . . . . . . . 9 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥))
14286, 141mpbird 248 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))𝑧𝑥)
143 supxrre 12364 . . . . . . . 8 ((ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))𝑧𝑥) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ))
144128, 136, 142, 143syl3anc 1490 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ))
145 rnco2 5830 . . . . . . . . 9 ran (vol ∘ (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
146 eqidd 2766 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
147 volf 23601 . . . . . . . . . . . . 13 vol:dom vol⟶(0[,]+∞)
148147a1i 11 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → vol:dom vol⟶(0[,]+∞))
149148feqmptd 6442 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → vol = (𝑦 ∈ dom vol ↦ (vol‘𝑦)))
150 fveq2 6379 . . . . . . . . . . 11 (𝑦 = ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) → (vol‘𝑦) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
15117, 146, 149, 150fmptco 6591 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol ∘ (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
152151rneqd 5523 . . . . . . . . 9 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (vol ∘ (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
153145, 152syl5reqr 2814 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
154153supeq1d 8563 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ))
155144, 154eqtr3d 2801 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ))
156105, 127, 1553eqtr4d 2809 . . . . 5 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ))
15787, 156breqtrrd 4839 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ⇝ (vol‘(𝐹 “ {𝑘})))
158 i1ff 23748 . . . . . . . 8 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
159 frn 6231 . . . . . . . 8 (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
1603, 158, 1593syl 18 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℝ)
161160ssdifssd 3912 . . . . . 6 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
162161sselda 3763 . . . . 5 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ)
163162recnd 10326 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ)
164 nnex 11285 . . . . . 6 ℕ ∈ V
165164mptex 6683 . . . . 5 (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) ∈ V
166165a1i 11 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) ∈ V)
16733ffvelrnda 6553 . . . . 5 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ∈ ℝ)
168167recnd 10326 . . . 4 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ∈ ℂ)
16954oveq2d 6862 . . . . . . 7 (𝑛 = 𝑗 → (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
170 eqid 2765 . . . . . . 7 (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) = (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
171 ovex 6878 . . . . . . 7 (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))) ∈ V
172169, 170, 171fvmpt 6475 . . . . . 6 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
17357oveq2d 6862 . . . . . 6 (𝑗 ∈ ℕ → (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗)) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
174172, 173eqtr4d 2802 . . . . 5 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗)))
175174adantl 473 . . . 4 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗)))
1761, 9, 157, 163, 166, 168, 175climmulc2 14666 . . 3 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) ⇝ (𝑘 · (vol‘(𝐹 “ {𝑘}))))
177164mptex 6683 . . . 4 (𝑛 ∈ ℕ ↦ (∫1𝐺)) ∈ V
178177a1i 11 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ∈ V)
179162adantr 472 . . . . . . . 8 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → 𝑘 ∈ ℝ)
180179, 32remulcld 10328 . . . . . . 7 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ∈ ℝ)
181180fmpttd 6579 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))):ℕ⟶ℝ)
182181ffvelrnda 6553 . . . . 5 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) ∈ ℝ)
183182recnd 10326 . . . 4 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) ∈ ℂ)
184183anasss 458 . . 3 ((𝜑 ∧ (𝑘 ∈ (ran 𝐹 ∖ {0}) ∧ 𝑗 ∈ ℕ)) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) ∈ ℂ)
1853adantr 472 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐹 ∈ dom ∫1)
186 itg1climres.5 . . . . . . . . . 10 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
187186i1fres 23777 . . . . . . . . 9 ((𝐹 ∈ dom ∫1 ∧ (𝐴𝑛) ∈ dom vol) → 𝐺 ∈ dom ∫1)
188185, 14, 187syl2anc 579 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝐺 ∈ dom ∫1)
1898adantr 472 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ∈ Fin)
190 ffn 6225 . . . . . . . . . . . . . 14 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
1913, 158, 1903syl 18 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℝ)
192191adantr 472 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐹 Fn ℝ)
193 fnfvelrn 6550 . . . . . . . . . . . 12 ((𝐹 Fn ℝ ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ran 𝐹)
194192, 193sylan 575 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ran 𝐹)
195 i1f0rn 23754 . . . . . . . . . . . . 13 (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹)
1963, 195syl 17 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ran 𝐹)
197196ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran 𝐹)
198194, 197ifcld 4290 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ∈ ran 𝐹)
199198, 186fmptd 6578 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℝ⟶ran 𝐹)
200 frn 6231 . . . . . . . . 9 (𝐺:ℝ⟶ran 𝐹 → ran 𝐺 ⊆ ran 𝐹)
201 ssdif 3909 . . . . . . . . 9 (ran 𝐺 ⊆ ran 𝐹 → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
202199, 200, 2013syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
203160adantr 472 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
204203ssdifd 3910 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))
205 itg1val2 23756 . . . . . . . 8 ((𝐺 ∈ dom ∫1 ∧ ((ran 𝐹 ∖ {0}) ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}) ∧ (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))) → (∫1𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐺 “ {𝑘}))))
206188, 189, 202, 204, 205syl13anc 1491 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫1𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐺 “ {𝑘}))))
207 fvex 6392 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑥) ∈ V
208 c0ex 10291 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
209207, 208ifex 4293 . . . . . . . . . . . . . . . . . . . 20 if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ∈ V
210186fvmpt2 6484 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ∈ V) → (𝐺𝑥) = if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
211209, 210mpan2 682 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ → (𝐺𝑥) = if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
212211adantl 473 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝐺𝑥) = if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
213212eqeq1d 2767 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑥) = 𝑘 ↔ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘))
214 eldifsni 4478 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0)
215214ad2antlr 718 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ≠ 0)
216 neeq1 2999 . . . . . . . . . . . . . . . . . . . 20 (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ≠ 0 ↔ 𝑘 ≠ 0))
217215, 216syl5ibrcom 238 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ≠ 0))
218 iffalse 4254 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ (𝐴𝑛) → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 0)
219218necon1ai 2964 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ≠ 0 → 𝑥 ∈ (𝐴𝑛))
220217, 219syl6 35 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘𝑥 ∈ (𝐴𝑛)))
221220pm4.71rd 558 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 ↔ (𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘)))
222213, 221bitrd 270 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑥) = 𝑘 ↔ (𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘)))
223 iftrue 4251 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴𝑛) → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = (𝐹𝑥))
224223eqeq1d 2767 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴𝑛) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 ↔ (𝐹𝑥) = 𝑘))
225224pm5.32i 570 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘) ↔ (𝑥 ∈ (𝐴𝑛) ∧ (𝐹𝑥) = 𝑘))
226 ancom 452 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝐴𝑛) ∧ (𝐹𝑥) = 𝑘) ↔ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛)))
227225, 226bitri 266 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘) ↔ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛)))
228222, 227syl6bb 278 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑥) = 𝑘 ↔ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛))))
229228pm5.32da 574 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘) ↔ (𝑥 ∈ ℝ ∧ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛)))))
230 anass 460 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛)) ↔ (𝑥 ∈ ℝ ∧ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛))))
231229, 230syl6bbr 280 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘) ↔ ((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛))))
232 i1ff 23748 . . . . . . . . . . . . . . . 16 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
233 ffn 6225 . . . . . . . . . . . . . . . 16 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
234188, 232, 2333syl 18 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝐺 Fn ℝ)
235234adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐺 Fn ℝ)
236 fniniseg 6532 . . . . . . . . . . . . . 14 (𝐺 Fn ℝ → (𝑥 ∈ (𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘)))
237235, 236syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘)))
238 elin 3960 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ↔ (𝑥 ∈ (𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴𝑛)))
239192adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ)
240 fniniseg 6532 . . . . . . . . . . . . . . . 16 (𝐹 Fn ℝ → (𝑥 ∈ (𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘)))
241239, 240syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘)))
242241anbi1d 623 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ (𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛))))
243238, 242syl5bb 274 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛))))
244231, 237, 2433bitr4d 302 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
245244alrimiv 2022 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑥(𝑥 ∈ (𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
246 nfmpt1 4908 . . . . . . . . . . . . . . 15 𝑥(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
247186, 246nfcxfr 2905 . . . . . . . . . . . . . 14 𝑥𝐺
248247nfcnv 5471 . . . . . . . . . . . . 13 𝑥𝐺
249 nfcv 2907 . . . . . . . . . . . . 13 𝑥{𝑘}
250248, 249nfima 5658 . . . . . . . . . . . 12 𝑥(𝐺 “ {𝑘})
251 nfcv 2907 . . . . . . . . . . . 12 𝑥((𝐹 “ {𝑘}) ∩ (𝐴𝑛))
252250, 251cleqf 2933 . . . . . . . . . . 11 ((𝐺 “ {𝑘}) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ↔ ∀𝑥(𝑥 ∈ (𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
253245, 252sylibr 225 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐺 “ {𝑘}) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))
254253fveq2d 6383 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐺 “ {𝑘})) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
255254oveq2d 6862 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(𝐺 “ {𝑘}))) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
256255sumeq2dv 14732 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐺 “ {𝑘}))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
257206, 256eqtrd 2799 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫1𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
258257mpteq2dva 4905 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))))
259258fveq1d 6381 . . . 4 (𝜑 → ((𝑛 ∈ ℕ ↦ (∫1𝐺))‘𝑗) = ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗))
260169sumeq2sdv 14734 . . . . . 6 (𝑛 = 𝑗 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
261 eqid 2765 . . . . . 6 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
262 sumex 14717 . . . . . 6 Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))) ∈ V
263260, 261, 262fvmpt 6475 . . . . 5 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
264172sumeq2sdv 14734 . . . . 5 (𝑗 ∈ ℕ → Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
265263, 264eqtr4d 2802 . . . 4 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗))
266259, 265sylan9eq 2819 . . 3 ((𝜑𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫1𝐺))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗))
2671, 2, 8, 176, 178, 184, 266climfsum 14850 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ⇝ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐹 “ {𝑘}))))
268 itg1val 23755 . . 3 (𝐹 ∈ dom ∫1 → (∫1𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐹 “ {𝑘}))))
2693, 268syl 17 . 2 (𝜑 → (∫1𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐹 “ {𝑘}))))
270267, 269breqtrrd 4839 1 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ⇝ (∫1𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3731  cin 3733  wss 3734  c0 4081  ifcif 4245  {csn 4336   cuni 4596   ciun 4678   class class class wbr 4811  cmpt 4890  ccnv 5278  dom cdm 5279  ran crn 5280  cima 5282  ccom 5283   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6846  Fincfn 8164  supcsup 8557  cc 10191  cr 10192  0cc0 10193  1c1 10194   + caddc 10196   · cmul 10198  +∞cpnf 10329  *cxr 10331   < clt 10332  cle 10333  cn 11278  [,]cicc 12385  cli 14514  Σcsu 14715  vol*covol 23534  volcvol 23535  1citg1 23687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cc 9514  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-disj 4780  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-z 11629  df-uz 11892  df-q 11995  df-rp 12034  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12386  df-ico 12388  df-icc 12389  df-fz 12539  df-fzo 12679  df-fl 12806  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-rlim 14519  df-sum 14716  df-rest 16363  df-topgen 16384  df-psmet 20025  df-xmet 20026  df-met 20027  df-bl 20028  df-mopn 20029  df-top 20992  df-topon 21009  df-bases 21044  df-cmp 21484  df-ovol 23536  df-vol 23537  df-mbf 23691  df-itg1 23692
This theorem is referenced by:  itg2monolem1  23822
  Copyright terms: Public domain W3C validator