Theorem itg1climres 25763

Description: Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is ℝ 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	⊢ (𝜑 → 𝐴:ℕ⟶dom vol)
itg1climres.2	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))
itg1climres.3	⊢ (𝜑 → ∪ ran 𝐴 = ℝ)
itg1climres.4	⊢ (𝜑 → 𝐹 ∈ dom ∫1)
itg1climres.5	⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))

Assertion

Ref	Expression
itg1climres	⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹))

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.

Step	Hyp	Ref	Expression
1		nnuz 12918	. . 3 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 12645	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		itg1climres.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1frn 25725	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
6		difss 4145	. . . 4 ⊢ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹
7		ssfi 9211	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin)
8	5, 6, 7	sylancl 586	. . 3 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin)
9		1zzd 12645	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 1 ∈ ℤ)
10		i1fima 25726	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {𝑘}) ∈ dom vol)
11	3, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐹 “ {𝑘}) ∈ dom vol)
12	11	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ {𝑘}) ∈ dom vol)
13		itg1climres.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:ℕ⟶dom vol)
14	13	ffvelcdmda 7103	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ dom vol)
15	14	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ dom vol)
16		inmbl 25590	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘𝑛) ∈ dom vol) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol)
17	12, 15, 16	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol)
18		mblvol 25578	. . . . . . . . 9 ⊢ (((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
19	17, 18	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
20		inss1 4244	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘})
21	20	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘}))
22		mblss 25579	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (◡𝐹 “ {𝑘}) ⊆ ℝ)
23	12, 22	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ {𝑘}) ⊆ ℝ)
24		mblvol 25578	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘})))
25	12, 24	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘})))
26		i1fima2sn 25728	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
27	3, 26	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
28	27	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
29	25, 28	eqeltrrd 2839	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘(◡𝐹 “ {𝑘})) ∈ ℝ)
30		ovolsscl 25534	. . . . . . . . 9 ⊢ ((((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘}) ∧ (◡𝐹 “ {𝑘}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑘})) ∈ ℝ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
31	21, 23, 29, 30	syl3anc 1370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
32	19, 31	eqeltrd 2838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
33	32	fmpttd 7134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))):ℕ⟶ℝ)
34		itg1climres.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))
35	34	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))
36		sslin 4250	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
38	13	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴:ℕ⟶dom vol)
39		peano2nn 12275	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40		ffvelcdm 7100	. . . . . . . . . . . . . 14 ⊢ ((𝐴:ℕ⟶dom vol ∧ (𝑛 + 1) ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
41	38, 39, 40	syl2an 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
42		inmbl 25590	. . . . . . . . . . . . 13 ⊢ (((◡𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘(𝑛 + 1)) ∈ dom vol) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
43	12, 41, 42	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
44		mblss 25579	. . . . . . . . . . . 12 ⊢ (((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ)
45	43, 44	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ)
46		ovolss 25533	. . . . . . . . . . 11 ⊢ ((((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∧ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
47	37, 45, 46	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
48		mblvol 25578	. . . . . . . . . . 11 ⊢ (((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
49	43, 48	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
50	47, 19, 49	3brtr4d 5179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
51	50	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
52		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗))
53	52	ineq2d 4227	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))
54	53	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))))
55		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
56		fvex 6919	. . . . . . . . . . . 12 ⊢ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ∈ V
57	54, 55, 56	fvmpt 7015	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))))
58		peano2nn 12275	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
59		fveq2 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → (𝐴‘𝑛) = (𝐴‘(𝑗 + 1)))
60	59	ineq2d 4227	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
61	60	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑗 + 1) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
62		fvex 6919	. . . . . . . . . . . . 13 ⊢ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))) ∈ V
63	61, 55, 62	fvmpt 7015	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
64	58, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
65	57, 64	breq12d 5160	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))))
66	65	ralbiia 3088	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
67		fvoveq1 7453	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑗 + 1)))
68	67	ineq2d 4227	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
69	68	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
70	54, 69	breq12d 5160	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → ((vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))))
71	70	cbvralvw 3234	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
72	66, 71	bitr4i 278	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) ↔ ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
73	51, 72	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)))
74	73	r19.21bi 3248	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)))
75		ovolss 25533	. . . . . . . . . . 11 ⊢ ((((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘}) ∧ (◡𝐹 “ {𝑘}) ⊆ ℝ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘(◡𝐹 “ {𝑘})))
76	20, 23, 75	sylancr 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘(◡𝐹 “ {𝑘})))
77	76, 19, 25	3brtr4d 5179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
78	77	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
79	57	breq1d 5157	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘}))))
80	79	ralbiia 3088	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘})))
81	54	breq1d 5157	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → ((vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘}))))
82	81	cbvralvw 3234	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘})))
83	80, 82	bitr4i 278	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
84	78, 83	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})))
85		brralrspcev 5207	. . . . . . 7 ⊢ (((vol‘(◡𝐹 “ {𝑘})) ∈ ℝ ∧ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘}))) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥)
86	27, 84, 85	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥)
87	1, 9, 33, 74, 86	climsup 15702	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
88	17	fmpttd 7134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol)
89	37	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
90		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))
91		fvex 6919	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑗) ∈ V
92	91	inex2 5323	. . . . . . . . . . . 12 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ∈ V
93	53, 90, 92	fvmpt 7015	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))
94		fvex 6919	. . . . . . . . . . . . . 14 ⊢ (𝐴‘(𝑗 + 1)) ∈ V
95	94	inex2 5323	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))) ∈ V
96	60, 90, 95	fvmpt 7015	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
97	58, 96	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
98	93, 97	sseq12d 4028	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) ↔ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
99	98	ralbiia 3088	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) ↔ ∀𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
100	53, 68	sseq12d 4028	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
101	100	cbvralvw 3234	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ∀𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
102	99, 101	bitr4i 278	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) ↔ ∀𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
103	89, 102	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)))
104		volsup 25604	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol ∧ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1))) → (vol‘∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
105	88, 103, 104	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
106	93	iuneq2i 5017	. . . . . . . . . 10 ⊢ ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ∪ 𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))
107	53	cbviunv 5044	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ∪ 𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))
108		iunin2 5075	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
109	106, 107, 108	3eqtr2i 2768	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
110		ffn 6736	. . . . . . . . . . . . . 14 ⊢ (𝐴:ℕ⟶dom vol → 𝐴 Fn ℕ)
111		fniunfv 7266	. . . . . . . . . . . . . 14 ⊢ (𝐴 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ∪ ran 𝐴)
112	13, 110, 111	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ∪ ran 𝐴)
113		itg1climres.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ ran 𝐴 = ℝ)
114	112, 113	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ℝ)
115	114	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ℝ)
116	115	ineq2d 4227	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ ℝ))
117	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ∈ dom vol)
118	117, 22	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ ℝ)
119		dfss2 3980	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑘}) ⊆ ℝ ↔ ((◡𝐹 “ {𝑘}) ∩ ℝ) = (◡𝐹 “ {𝑘}))
120	118, 119	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑘}) ∩ ℝ) = (◡𝐹 “ {𝑘}))
121	116, 120	eqtrd 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = (◡𝐹 “ {𝑘}))
122	109, 121	eqtrid 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = (◡𝐹 “ {𝑘}))
123		ffn 6736	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol → (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) Fn ℕ)
124		fniunfv 7266	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) Fn ℕ → ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
125	88, 123, 124	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
126	122, 125	eqtr3d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) = ∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
127	126	fveq2d 6910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = (vol‘∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
128	33	frnd 6744	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⊆ ℝ)
129	33	fdmd 6746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ℕ)
130		1nn 12274	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
131		ne0i 4346	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
132	130, 131	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ℕ ≠ ∅)
133	129, 132	eqnetrd 3005	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
134		dm0rn0 5937	. . . . . . . . . 10 ⊢ (dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ∅)
135	134	necon3bii 2990	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
136	133, 135	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
137		ffn 6736	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) Fn ℕ)
138		breq1 5150	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) → (𝑧 ≤ 𝑥 ↔ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
139	138	ralrn 7107	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
140	33, 137, 139	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
141	140	rexbidv 3176	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
142	86, 141	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥)
143		supxrre 13365	. . . . . . . 8 ⊢ ((ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
144	128, 136, 142, 143	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
145		volf 25577	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
146	145	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → vol:dom vol⟶(0[,]+∞))
147	146, 17	cofmpt 7151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
148	147	rneqd 5951	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
149		rnco2 6274	. . . . . . . . 9 ⊢ ran (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
150	148, 149	eqtr3di 2789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
151	150	supeq1d 9483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
152	144, 151	eqtr3d 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
153	105, 127, 152	3eqtr4d 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
154	87, 153	breqtrrd 5175	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⇝ (vol‘(◡𝐹 “ {𝑘})))
155		i1ff 25724	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
156		frn 6743	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
157	3, 155, 156	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
158	157	ssdifssd 4156	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
159	158	sselda 3994	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ)
160	159	recnd 11286	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ)
161		nnex 12269	. . . . . 6 ⊢ ℕ ∈ V
162	161	mptex 7242	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) ∈ V
163	162	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) ∈ V)
164	33	ffvelcdmda 7103	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ∈ ℝ)
165	164	recnd 11286	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ∈ ℂ)
166	54	oveq2d 7446	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
167		eqid 2734	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) = (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
168		ovex 7463	. . . . . . 7 ⊢ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))) ∈ V
169	166, 167, 168	fvmpt 7015	. . . . . 6 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
170	57	oveq2d 7446	. . . . . 6 ⊢ (𝑗 ∈ ℕ → (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗)) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
171	169, 170	eqtr4d 2777	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗)))
172	171	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗)))
173	1, 9, 154, 160, 163, 165, 172	climmulc2 15669	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) ⇝ (𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
174	161	mptex 7242	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ∈ V
175	174	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ∈ V)
176	159	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → 𝑘 ∈ ℝ)
177	176, 32	remulcld 11288	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ∈ ℝ)
178	177	fmpttd 7134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))):ℕ⟶ℝ)
179	178	ffvelcdmda 7103	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℝ)
180	179	recnd 11286	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℂ)
181	180	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (ran 𝐹 ∖ {0}) ∧ 𝑗 ∈ ℕ)) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℂ)
182	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ dom ∫1)
183		itg1climres.5	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
184	183	i1fres 25754	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫1 ∧ (𝐴‘𝑛) ∈ dom vol) → 𝐺 ∈ dom ∫1)
185	182, 14, 184	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ dom ∫1)
186	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ∈ Fin)
187		ffn 6736	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
188	3, 155, 187	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℝ)
189	188	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 Fn ℝ)
190		fnfvelrn 7099	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran 𝐹)
191	189, 190	sylan 580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran 𝐹)
192		i1f0rn 25730	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹)
193	3, 192	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ran 𝐹)
194	193	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran 𝐹)
195	191, 194	ifcld 4576	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ ran 𝐹)
196	195, 183	fmptd 7133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℝ⟶ran 𝐹)
197		frn 6743	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ran 𝐹 → ran 𝐺 ⊆ ran 𝐹)
198		ssdif 4153	. . . . . . . . 9 ⊢ (ran 𝐺 ⊆ ran 𝐹 → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
199	196, 197, 198	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
200	157	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
201	200	ssdifd 4154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))
202		itg1val2 25732	. . . . . . . 8 ⊢ ((𝐺 ∈ dom ∫1 ∧ ((ran 𝐹 ∖ {0}) ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}) ∧ (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))) → (∫1‘𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐺 “ {𝑘}))))
203	185, 186, 199, 201, 202	syl13anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫1‘𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐺 “ {𝑘}))))
204		fvex 6919	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘𝑥) ∈ V
205		c0ex 11252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
206	204, 205	ifex 4580	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ V
207	183	fvmpt2 7026	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ V) → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
208	206, 207	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
209	208	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
210	209	eqeq1d 2736	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘))
211		eldifsni 4794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0)
212	211	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ≠ 0)
213		neeq1 3000	. . . . . . . . . . . . . . . . . . . 20 ⊢ (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ≠ 0 ↔ 𝑘 ≠ 0))
214	212, 213	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ≠ 0))
215		iffalse 4539	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 ∈ (𝐴‘𝑛) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 0)
216	215	necon1ai 2965	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ≠ 0 → 𝑥 ∈ (𝐴‘𝑛))
217	214, 216	syl6 35	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 → 𝑥 ∈ (𝐴‘𝑛)))
218	217	pm4.71rd 562	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 ↔ (𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘)))
219	210, 218	bitrd 279	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ (𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘)))
220		iftrue 4536	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴‘𝑛) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = (𝐹‘𝑥))
221	220	eqeq1d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴‘𝑛) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 ↔ (𝐹‘𝑥) = 𝑘))
222	221	pm5.32i 574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘) ↔ (𝑥 ∈ (𝐴‘𝑛) ∧ (𝐹‘𝑥) = 𝑘))
223	222	biancomi 462	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘) ↔ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛)))
224	219, 223	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛))))
225	224	pm5.32da 579	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘) ↔ (𝑥 ∈ ℝ ∧ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛)))))
226		anass 468	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛)) ↔ (𝑥 ∈ ℝ ∧ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛))))
227	225, 226	bitr4di 289	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘) ↔ ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛))))
228		i1ff 25724	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
229		ffn 6736	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
230	185, 228, 229	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 Fn ℝ)
231	230	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐺 Fn ℝ)
232		fniniseg 7079	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn ℝ → (𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘)))
233	231, 232	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘)))
234		elin 3978	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ (𝑥 ∈ (◡𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴‘𝑛)))
235	189	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ)
236		fniniseg 7079	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)))
237	235, 236	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)))
238	237	anbi1d 631	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ (◡𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴‘𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛))))
239	234, 238	bitrid 283	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛))))
240	227, 233, 239	3bitr4d 311	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
241	240	alrimiv 1924	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑥(𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
242		nfmpt1 5255	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
243	183, 242	nfcxfr 2900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐺
244	243	nfcnv 5891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥◡𝐺
245		nfcv 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑘}
246	244, 245	nfima 6087	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(◡𝐺 “ {𝑘})
247		nfcv 2902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))
248	246, 247	cleqf 2931	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑘}) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ ∀𝑥(𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
249	241, 248	sylibr 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐺 “ {𝑘}) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))
250	249	fveq2d 6910	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐺 “ {𝑘})) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
251	250	oveq2d 7446	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐺 “ {𝑘}))) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
252	251	sumeq2dv 15734	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐺 “ {𝑘}))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
253	203, 252	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫1‘𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
254	253	mpteq2dva 5247	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))))
255	254	fveq1d 6908	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (∫1‘𝐺))‘𝑗) = ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗))
256	166	sumeq2sdv 15735	. . . . . 6 ⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
257		eqid 2734	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
258		sumex 15720	. . . . . 6 ⊢ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))) ∈ V
259	256, 257, 258	fvmpt 7015	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
260	169	sumeq2sdv 15735	. . . . 5 ⊢ (𝑗 ∈ ℕ → Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
261	259, 260	eqtr4d 2777	. . . 4 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗))
262	255, 261	sylan9eq 2794	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫1‘𝐺))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗))
263	1, 2, 8, 173, 175, 181, 262	climfsum 15852	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
264		itg1val 25731	. . 3 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
265	3, 264	syl 17	. 2 ⊢ (𝜑 → (∫1‘𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
266	263, 265	breqtrrd 5175	1 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1534   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3477   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338  ifcif 4530  {csn 4630  ∪ cuni 4911  ∪ ciun 4995   class class class wbr 5147   ↦ cmpt 5230  ^◡ccnv 5687  dom cdm 5688  ran crn 5689   “ cima 5691   ∘ ccom 5692   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  Fincfn 8983  supcsup 9477  ℂcc 11150  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  +∞cpnf 11289  ℝ^*cxr 11291   < clt 11292   ≤ cle 11293  ℕcn 12263  [,]cicc 13386   ⇝ cli 15516  Σcsu 15718  vol*covol 25510  volcvol 25511  ∫₁citg1 25663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-rest 17468  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-top 22915  df-topon 22932  df-bases 22968  df-cmp 23410  df-ovol 25512  df-vol 25513  df-mbf 25667  df-itg1 25668

This theorem is referenced by:  itg2monolem1  25799