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Theorem itg1climres 24318
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 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 12284 . . 3 ℕ = (ℤ‘1)
2 1zzd 12016 . . 3 (𝜑 → 1 ∈ ℤ)
3 itg1climres.4 . . . . 5 (𝜑𝐹 ∈ dom ∫1)
4 i1frn 24281 . . . . 5 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
53, 4syl 17 . . . 4 (𝜑 → ran 𝐹 ∈ Fin)
6 difss 4111 . . . 4 (ran 𝐹 ∖ {0}) ⊆ ran 𝐹
7 ssfi 8741 . . . 4 ((ran 𝐹 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin)
85, 6, 7sylancl 588 . . 3 (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin)
9 1zzd 12016 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 1 ∈ ℤ)
10 i1fima 24282 . . . . . . . . . . . 12 (𝐹 ∈ dom ∫1 → (𝐹 “ {𝑘}) ∈ dom vol)
113, 10syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ {𝑘}) ∈ dom vol)
1211ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐹 “ {𝑘}) ∈ dom vol)
13 itg1climres.1 . . . . . . . . . . . 12 (𝜑𝐴:ℕ⟶dom vol)
1413ffvelrnda 6854 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ dom vol)
1514adantlr 713 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ dom vol)
16 inmbl 24146 . . . . . . . . . 10 (((𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴𝑛) ∈ dom vol) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ∈ dom vol)
1712, 15, 16syl2anc 586 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ∈ dom vol)
18 mblvol 24134 . . . . . . . . 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 4208 . . . . . . . . . 10 ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘})
2120a1i 11 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘}))
22 mblss 24135 . . . . . . . . . 10 ((𝐹 “ {𝑘}) ∈ dom vol → (𝐹 “ {𝑘}) ⊆ ℝ)
2312, 22syl 17 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐹 “ {𝑘}) ⊆ ℝ)
24 mblvol 24134 . . . . . . . . . . 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 24284 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) ∈ ℝ)
273, 26sylan 582 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) ∈ ℝ)
2827adantr 483 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹 “ {𝑘})) ∈ ℝ)
2925, 28eqeltrrd 2917 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝐹 “ {𝑘})) ∈ ℝ)
30 ovolsscl 24090 . . . . . . . . 9 ((((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘}) ∧ (𝐹 “ {𝑘}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑘})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ∈ ℝ)
3121, 23, 29, 30syl3anc 1367 . . . . . . . 8 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ∈ ℝ)
3219, 31eqeltrd 2916 . . . . . . 7 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ∈ ℝ)
3332fmpttd 6882 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))):ℕ⟶ℝ)
34 itg1climres.2 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)))
3534adantlr 713 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)))
36 sslin 4214 . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴:ℕ⟶dom vol)
39 peano2nn 11653 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40 ffvelrn 6852 . . . . . . . . . . . . . 14 ((𝐴:ℕ⟶dom vol ∧ (𝑛 + 1) ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
4138, 39, 40syl2an 597 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
42 inmbl 24146 . . . . . . . . . . . . 13 (((𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘(𝑛 + 1)) ∈ dom vol) → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
4312, 41, 42syl2anc 586 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
44 mblss 24135 . . . . . . . . . . . 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 24089 . . . . . . . . . . 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 586 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
48 mblvol 24134 . . . . . . . . . . 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 5101 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
5150ralrimiva 3185 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
52 fveq2 6673 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
5352ineq2d 4192 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))
5453fveq2d 6677 . . . . . . . . . . . 12 (𝑛 = 𝑗 → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))))
55 eqid 2824 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
56 fvex 6686 . . . . . . . . . . . 12 (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ∈ V
5754, 55, 56fvmpt 6771 . . . . . . . . . . 11 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))))
58 peano2nn 11653 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
59 fveq2 6673 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (𝐴𝑛) = (𝐴‘(𝑗 + 1)))
6059ineq2d 4192 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
6160fveq2d 6677 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
62 fvex 6686 . . . . . . . . . . . . 13 (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))) ∈ V
6361, 55, 62fvmpt 6771 . . . . . . . . . . . 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 5082 . . . . . . . . . 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 3167 . . . . . . . . 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 7182 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑗 + 1)))
6867ineq2d 4192 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) = ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
6968fveq2d 6677 . . . . . . . . . . 11 (𝑛 = 𝑗 → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
7054, 69breq12d 5082 . . . . . . . . . 10 (𝑛 = 𝑗 → ((vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) ↔ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))))
7170cbvralvw 3452 . . . . . . . . 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 280 . . . . . . . 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 236 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)))
7473r19.21bi 3211 . . . . . 6 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘(𝑗 + 1)))
75 ovolss 24089 . . . . . . . . . . 11 ((((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ (𝐹 “ {𝑘}) ∧ (𝐹 “ {𝑘}) ⊆ ℝ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol*‘(𝐹 “ {𝑘})))
7620, 23, 75sylancr 589 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol*‘(𝐹 “ {𝑘})))
7776, 19, 253brtr4d 5101 . . . . . . . . 9 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})))
7877ralrimiva 3185 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})))
7957breq1d 5079 . . . . . . . . . 10 (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})) ↔ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘}))))
8079ralbiia 3167 . . . . . . . . 9 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘})))
8154breq1d 5079 . . . . . . . . . 10 (𝑛 = 𝑗 → ((vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})) ↔ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘}))))
8281cbvralvw 3452 . . . . . . . . 9 (∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗))) ≤ (vol‘(𝐹 “ {𝑘})))
8380, 82bitr4i 280 . . . . . . . 8 (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})) ↔ ∀𝑛 ∈ ℕ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) ≤ (vol‘(𝐹 “ {𝑘})))
8478, 83sylibr 236 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ (vol‘(𝐹 “ {𝑘})))
85 brralrspcev 5129 . . . . . . 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 586 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥)
871, 9, 33, 74, 86climsup 15029 . . . . 5 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ))
8817fmpttd 6882 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))):ℕ⟶dom vol)
8937ralrimiva 3185 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
90 eqid 2824 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))
91 fvex 6686 . . . . . . . . . . . . 13 (𝐴𝑗) ∈ V
9291inex2 5225 . . . . . . . . . . . 12 ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ∈ V
9353, 90, 92fvmpt 6771 . . . . . . . . . . 11 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))
94 fvex 6686 . . . . . . . . . . . . . 14 (𝐴‘(𝑗 + 1)) ∈ V
9594inex2 5225 . . . . . . . . . . . . 13 ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))) ∈ V
9660, 90, 95fvmpt 6771 . . . . . . . . . . . 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 4003 . . . . . . . . . 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 3167 . . . . . . . . 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 4003 . . . . . . . . . 10 (𝑛 = 𝑗 → (((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
101100cbvralvw 3452 . . . . . . . . 9 (∀𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ∀𝑗 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗)) ⊆ ((𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
10299, 101bitr4i 280 . . . . . . . 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 236 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘(𝑗 + 1)))
104 volsup 24160 . . . . . . 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 586 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ))
10693iuneq2i 4943 . . . . . . . . . 10 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = 𝑗 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗))
10753cbviunv 4968 . . . . . . . . . 10 𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = 𝑗 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑗))
108 iunin2 4996 . . . . . . . . . 10 𝑛 ∈ ℕ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛))
109106, 107, 1083eqtr2i 2853 . . . . . . . . 9 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛))
110 ffn 6517 . . . . . . . . . . . . . 14 (𝐴:ℕ⟶dom vol → 𝐴 Fn ℕ)
111 fniunfv 7009 . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . 12 (𝜑 𝑛 ∈ ℕ (𝐴𝑛) = ℝ)
115114adantr 483 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑛 ∈ ℕ (𝐴𝑛) = ℝ)
116115ineq2d 4192 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛)) = ((𝐹 “ {𝑘}) ∩ ℝ))
11711adantr 483 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑘}) ∈ dom vol)
118117, 22syl 17 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑘}) ⊆ ℝ)
119 df-ss 3955 . . . . . . . . . . 11 ((𝐹 “ {𝑘}) ⊆ ℝ ↔ ((𝐹 “ {𝑘}) ∩ ℝ) = (𝐹 “ {𝑘}))
120118, 119sylib 220 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑘}) ∩ ℝ) = (𝐹 “ {𝑘}))
121116, 120eqtrd 2859 . . . . . . . . 9 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑘}) ∩ 𝑛 ∈ ℕ (𝐴𝑛)) = (𝐹 “ {𝑘}))
122109, 121syl5eq 2871 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))‘𝑗) = (𝐹 “ {𝑘}))
123 ffn 6517 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))):ℕ⟶dom vol → (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))) Fn ℕ)
124 fniunfv 7009 . . . . . . . . 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 2861 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑘}) = ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
127126fveq2d 6677 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) = (vol‘ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
12833frnd 6524 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ⊆ ℝ)
12933fdmd 6526 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ℕ)
130 1nn 11652 . . . . . . . . . . 11 1 ∈ ℕ
131 ne0i 4303 . . . . . . . . . . 11 (1 ∈ ℕ → ℕ ≠ ∅)
132130, 131mp1i 13 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ℕ ≠ ∅)
133129, 132eqnetrd 3086 . . . . . . . . 9 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅)
134 dm0rn0 5798 . . . . . . . . . 10 (dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ∅)
135134necon3bii 3071 . . . . . . . . 9 (dom (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅)
136133, 135sylib 220 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ≠ ∅)
137 ffn 6517 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) Fn ℕ)
138 breq1 5072 . . . . . . . . . . . 12 (𝑧 = ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) → (𝑧𝑥 ↔ ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ≤ 𝑥))
139138ralrn 6857 . . . . . . . . . . 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 3300 . . . . . . . . 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 259 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))𝑧𝑥)
143 supxrre 12723 . . . . . . . 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 1367 . . . . . . 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 6109 . . . . . . . . 9 ran (vol ∘ (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
146 volf 24133 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
147146a1i 11 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → vol:dom vol⟶(0[,]+∞))
148147, 17cofmpt 6897 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol ∘ (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
149148rneqd 5811 . . . . . . . . 9 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (vol ∘ (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
150145, 149syl5reqr 2874 . . . . . . . 8 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
151150supeq1d 8913 . . . . . . 7 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ))
152144, 151eqtr3d 2861 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ*, < ))
153105, 127, 1523eqtr4d 2869 . . . . 5 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑘})) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))), ℝ, < ))
15487, 153breqtrrd 5097 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ⇝ (vol‘(𝐹 “ {𝑘})))
155 i1ff 24280 . . . . . . . 8 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
156 frn 6523 . . . . . . . 8 (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
1573, 155, 1563syl 18 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℝ)
158157ssdifssd 4122 . . . . . 6 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
159158sselda 3970 . . . . 5 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ)
160159recnd 10672 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ)
161 nnex 11647 . . . . . 6 ℕ ∈ V
162161mptex 6989 . . . . 5 (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) ∈ V
163162a1i 11 . . . 4 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) ∈ V)
16433ffvelrnda 6854 . . . . 5 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ∈ ℝ)
165164recnd 10672 . . . 4 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗) ∈ ℂ)
16654oveq2d 7175 . . . . . . 7 (𝑛 = 𝑗 → (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
167 eqid 2824 . . . . . . 7 (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) = (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
168 ovex 7192 . . . . . . 7 (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))) ∈ V
169166, 167, 168fvmpt 6771 . . . . . 6 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
17057oveq2d 7175 . . . . . 6 (𝑗 ∈ ℕ → (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗)) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
171169, 170eqtr4d 2862 . . . . 5 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗)))
172171adantl 484 . . . 4 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))‘𝑗)))
1731, 9, 154, 160, 163, 165, 172climmulc2 14996 . . 3 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) ⇝ (𝑘 · (vol‘(𝐹 “ {𝑘}))))
174161mptex 6989 . . . 4 (𝑛 ∈ ℕ ↦ (∫1𝐺)) ∈ V
175174a1i 11 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ∈ V)
176159adantr 483 . . . . . . . 8 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → 𝑘 ∈ ℝ)
177176, 32remulcld 10674 . . . . . . 7 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) ∈ ℝ)
178177fmpttd 6882 . . . . . 6 ((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))):ℕ⟶ℝ)
179178ffvelrnda 6854 . . . . 5 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) ∈ ℝ)
180179recnd 10672 . . . 4 (((𝜑𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) ∈ ℂ)
181180anasss 469 . . 3 ((𝜑 ∧ (𝑘 ∈ (ran 𝐹 ∖ {0}) ∧ 𝑗 ∈ ℕ)) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) ∈ ℂ)
1823adantr 483 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐹 ∈ dom ∫1)
183 itg1climres.5 . . . . . . . . . 10 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
184183i1fres 24309 . . . . . . . . 9 ((𝐹 ∈ dom ∫1 ∧ (𝐴𝑛) ∈ dom vol) → 𝐺 ∈ dom ∫1)
185182, 14, 184syl2anc 586 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝐺 ∈ dom ∫1)
1868adantr 483 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ∈ Fin)
187 ffn 6517 . . . . . . . . . . . . . 14 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
1883, 155, 1873syl 18 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℝ)
189188adantr 483 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐹 Fn ℝ)
190 fnfvelrn 6851 . . . . . . . . . . . 12 ((𝐹 Fn ℝ ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ran 𝐹)
191189, 190sylan 582 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ran 𝐹)
192 i1f0rn 24286 . . . . . . . . . . . . 13 (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹)
1933, 192syl 17 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ran 𝐹)
194193ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran 𝐹)
195191, 194ifcld 4515 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ∈ ran 𝐹)
196195, 183fmptd 6881 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℝ⟶ran 𝐹)
197 frn 6523 . . . . . . . . 9 (𝐺:ℝ⟶ran 𝐹 → ran 𝐺 ⊆ ran 𝐹)
198 ssdif 4119 . . . . . . . . 9 (ran 𝐺 ⊆ ran 𝐹 → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
199196, 197, 1983syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
200157adantr 483 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
201200ssdifd 4120 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))
202 itg1val2 24288 . . . . . . . 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}))))
203185, 186, 199, 201, 202syl13anc 1368 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫1𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐺 “ {𝑘}))))
204 fvex 6686 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑥) ∈ V
205 c0ex 10638 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
206204, 205ifex 4518 . . . . . . . . . . . . . . . . . . . 20 if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ∈ V
207183fvmpt2 6782 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ∈ V) → (𝐺𝑥) = if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
208206, 207mpan2 689 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ → (𝐺𝑥) = if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
209208adantl 484 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝐺𝑥) = if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
210209eqeq1d 2826 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑥) = 𝑘 ↔ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘))
211 eldifsni 4725 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0)
212211ad2antlr 725 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ≠ 0)
213 neeq1 3081 . . . . . . . . . . . . . . . . . . . 20 (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ≠ 0 ↔ 𝑘 ≠ 0))
214212, 213syl5ibrcom 249 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ≠ 0))
215 iffalse 4479 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ (𝐴𝑛) → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 0)
216215necon1ai 3046 . . . . . . . . . . . . . . . . . . 19 (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) ≠ 0 → 𝑥 ∈ (𝐴𝑛))
217214, 216syl6 35 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘𝑥 ∈ (𝐴𝑛)))
218217pm4.71rd 565 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 ↔ (𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘)))
219210, 218bitrd 281 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑥) = 𝑘 ↔ (𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘)))
220 iftrue 4476 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴𝑛) → if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = (𝐹𝑥))
221220eqeq1d 2826 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴𝑛) → (if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘 ↔ (𝐹𝑥) = 𝑘))
222221pm5.32i 577 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘) ↔ (𝑥 ∈ (𝐴𝑛) ∧ (𝐹𝑥) = 𝑘))
223222biancomi 465 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝐴𝑛) ∧ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0) = 𝑘) ↔ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛)))
224219, 223syl6bb 289 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑥) = 𝑘 ↔ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛))))
225224pm5.32da 581 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘) ↔ (𝑥 ∈ ℝ ∧ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛)))))
226 anass 471 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛)) ↔ (𝑥 ∈ ℝ ∧ ((𝐹𝑥) = 𝑘𝑥 ∈ (𝐴𝑛))))
227225, 226syl6bbr 291 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘) ↔ ((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛))))
228 i1ff 24280 . . . . . . . . . . . . . . . 16 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
229 ffn 6517 . . . . . . . . . . . . . . . 16 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
230185, 228, 2293syl 18 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝐺 Fn ℝ)
231230adantr 483 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐺 Fn ℝ)
232 fniniseg 6833 . . . . . . . . . . . . . 14 (𝐺 Fn ℝ → (𝑥 ∈ (𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘)))
233231, 232syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺𝑥) = 𝑘)))
234 elin 4172 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ↔ (𝑥 ∈ (𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴𝑛)))
235189adantr 483 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ)
236 fniniseg 6833 . . . . . . . . . . . . . . . 16 (𝐹 Fn ℝ → (𝑥 ∈ (𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘)))
237235, 236syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘)))
238237anbi1d 631 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ (𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛))))
239234, 238syl5bb 285 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴𝑛))))
240227, 233, 2393bitr4d 313 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
241240alrimiv 1927 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑥(𝑥 ∈ (𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
242 nfmpt1 5167 . . . . . . . . . . . . . . 15 𝑥(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))
243183, 242nfcxfr 2978 . . . . . . . . . . . . . 14 𝑥𝐺
244243nfcnv 5752 . . . . . . . . . . . . 13 𝑥𝐺
245 nfcv 2980 . . . . . . . . . . . . 13 𝑥{𝑘}
246244, 245nfima 5940 . . . . . . . . . . . 12 𝑥(𝐺 “ {𝑘})
247 nfcv 2980 . . . . . . . . . . . 12 𝑥((𝐹 “ {𝑘}) ∩ (𝐴𝑛))
248246, 247cleqf 3013 . . . . . . . . . . 11 ((𝐺 “ {𝑘}) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)) ↔ ∀𝑥(𝑥 ∈ (𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
249241, 248sylibr 236 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝐺 “ {𝑘}) = ((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))
250249fveq2d 6677 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐺 “ {𝑘})) = (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))
251250oveq2d 7175 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(𝐺 “ {𝑘}))) = (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
252251sumeq2dv 15063 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐺 “ {𝑘}))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
253203, 252eqtrd 2859 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫1𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
254253mpteq2dva 5164 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))))
255254fveq1d 6675 . . . 4 (𝜑 → ((𝑛 ∈ ℕ ↦ (∫1𝐺))‘𝑗) = ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗))
256166sumeq2sdv 15064 . . . . . 6 (𝑛 = 𝑗 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
257 eqid 2824 . . . . . 6 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))
258 sumex 15047 . . . . . 6 Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))) ∈ V
259256, 257, 258fvmpt 6771 . . . . 5 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
260169sumeq2sdv 15064 . . . . 5 (𝑗 ∈ ℕ → Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑗)))))
261259, 260eqtr4d 2862 . . . 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))
262255, 261sylan9eq 2879 . . 3 ((𝜑𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫1𝐺))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((𝐹 “ {𝑘}) ∩ (𝐴𝑛)))))‘𝑗))
2631, 2, 8, 173, 175, 181, 262climfsum 15178 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ⇝ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐹 “ {𝑘}))))
264 itg1val 24287 . . 3 (𝐹 ∈ dom ∫1 → (∫1𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐹 “ {𝑘}))))
2653, 264syl 17 . 2 (𝜑 → (∫1𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(𝐹 “ {𝑘}))))
266263, 265breqtrrd 5097 1 (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ⇝ (∫1𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1534   = wceq 1536  wcel 2113  wne 3019  wral 3141  wrex 3142  Vcvv 3497  cdif 3936  cin 3938  wss 3939  c0 4294  ifcif 4470  {csn 4570   cuni 4841   ciun 4922   class class class wbr 5069  cmpt 5149  ccnv 5557  dom cdm 5558  ran crn 5559  cima 5561  ccom 5562   Fn wfn 6353  wf 6354  cfv 6358  (class class class)co 7159  Fincfn 8512  supcsup 8907  cc 10538  cr 10539  0cc0 10540  1c1 10541   + caddc 10543   · cmul 10545  +∞cpnf 10675  *cxr 10677   < clt 10678  cle 10679  cn 11641  [,]cicc 12744  cli 14844  Σcsu 15045  vol*covol 24066  volcvol 24067  1citg1 24219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cc 9860  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-disj 5035  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fi 8878  df-sup 8909  df-inf 8910  df-oi 8977  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-rlim 14849  df-sum 15046  df-rest 16699  df-topgen 16720  df-psmet 20540  df-xmet 20541  df-met 20542  df-bl 20543  df-mopn 20544  df-top 21505  df-topon 21522  df-bases 21557  df-cmp 21998  df-ovol 24068  df-vol 24069  df-mbf 24223  df-itg1 24224
This theorem is referenced by:  itg2monolem1  24354
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