Theorem itg1climres 25699

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 12818	. . 3 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 12549	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		itg1climres.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1frn 25662	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
6		difss 4066	. . . 4 ⊢ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹
7		ssfi 9097	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin)
8	5, 6, 7	sylancl 592	. . 3 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin)
9		1zzd 12549	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 1 ∈ ℤ)
10		i1fima 25663	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {𝑘}) ∈ dom vol)
11	3, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐹 “ {𝑘}) ∈ dom vol)
12	11	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ {𝑘}) ∈ dom vol)
13		itg1climres.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:ℕ⟶dom vol)
14	13	ffvelcdmda 7025	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ dom vol)
15	14	adantlr 721	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ dom vol)
16		inmbl 25527	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘𝑛) ∈ dom vol) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol)
17	12, 15, 16	syl2anc 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol)
18		mblvol 25515	. . . . . . . . 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 4165	. . . . . . . . . 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 25516	. . . . . . . . . 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 25515	. . . . . . . . . . 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 25665	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
27	3, 26	sylan 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
28	27	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
29	25, 28	eqeltrrd 2840	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘(◡𝐹 “ {𝑘})) ∈ ℝ)
30		ovolsscl 25471	. . . . . . . . 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 1379	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
32	19, 31	eqeltrd 2839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
33	32	fmpttd 7056	. . . . . 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 721	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))
36		sslin 4171	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴:ℕ⟶dom vol)
39		peano2nn 12177	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40		ffvelcdm 7022	. . . . . . . . . . . . . 14 ⊢ ((𝐴:ℕ⟶dom vol ∧ (𝑛 + 1) ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
41	38, 39, 40	syl2an 602	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
42		inmbl 25527	. . . . . . . . . . . . 13 ⊢ (((◡𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘(𝑛 + 1)) ∈ dom vol) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
43	12, 41, 42	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
44		mblss 25516	. . . . . . . . . . . 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 25470	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
48		mblvol 25515	. . . . . . . . . . 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 5104	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
51	50	ralrimiva 3131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
52		fveq2 6827	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗))
53	52	ineq2d 4149	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))
54	53	fveq2d 6831	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))))
55		eqid 2739	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
56		fvex 6840	. . . . . . . . . . . 12 ⊢ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ∈ V
57	54, 55, 56	fvmpt 6935	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))))
58		peano2nn 12177	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
59		fveq2 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → (𝐴‘𝑛) = (𝐴‘(𝑗 + 1)))
60	59	ineq2d 4149	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
61	60	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑗 + 1) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
62		fvex 6840	. . . . . . . . . . . . 13 ⊢ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))) ∈ V
63	61, 55, 62	fvmpt 6935	. . . . . . . . . . . 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 5085	. . . . . . . . . 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 3083	. . . . . . . . 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 7379	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑗 + 1)))
68	67	ineq2d 4149	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
69	68	fveq2d 6831	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
70	54, 69	breq12d 5085	. . . . . . . . . 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 3217	. . . . . . . . 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 279	. . . . . . . 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 235	. . . . . . 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 3231	. . . . . 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 25470	. . . . . . . . . . 11 ⊢ ((((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘}) ∧ (◡𝐹 “ {𝑘}) ⊆ ℝ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘(◡𝐹 “ {𝑘})))
76	20, 23, 75	sylancr 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘(◡𝐹 “ {𝑘})))
77	76, 19, 25	3brtr4d 5104	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
78	77	ralrimiva 3131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
79	57	breq1d 5082	. . . . . . . . . 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 3083	. . . . . . . . 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 5082	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → ((vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘}))))
82	81	cbvralvw 3217	. . . . . . . . 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 279	. . . . . . . 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 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})))
85		brralrspcev 5132	. . . . . . 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 590	. . . . . 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 15623	. . . . 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 7056	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol)
89	37	ralrimiva 3131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
90		eqid 2739	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))
91		fvex 6840	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑗) ∈ V
92	91	inex2 5246	. . . . . . . . . . . 12 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ∈ V
93	53, 90, 92	fvmpt 6935	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))
94		fvex 6840	. . . . . . . . . . . . . 14 ⊢ (𝐴‘(𝑗 + 1)) ∈ V
95	94	inex2 5246	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))) ∈ V
96	60, 90, 95	fvmpt 6935	. . . . . . . . . . . 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 3948	. . . . . . . . . 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 3083	. . . . . . . . 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 3948	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
101	100	cbvralvw 3217	. . . . . . . . 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 279	. . . . . . . 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 235	. . . . . . 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 25541	. . . . . . 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 590	. . . . . 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 4943	. . . . . . . . . 10 ⊢ ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ∪ 𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))
107	53	cbviunv 4968	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ∪ 𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))
108		iunin2 5000	. . . . . . . . . 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 6655	. . . . . . . . . . . . . 14 ⊢ (𝐴:ℕ⟶dom vol → 𝐴 Fn ℕ)
111		fniunfv 7191	. . . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ℝ)
116	115	ineq2d 4149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ ℝ))
117	11	adantr 481	. . . . . . . . . . . 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 3901	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑘}) ⊆ ℝ ↔ ((◡𝐹 “ {𝑘}) ∩ ℝ) = (◡𝐹 “ {𝑘}))
120	118, 119	sylib 219	. . . . . . . . . 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 6655	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol → (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) Fn ℕ)
124		fniunfv 7191	. . . . . . . . 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 6831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = (vol‘∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
128	33	frnd 6663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⊆ ℝ)
129	33	fdmd 6665	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ℕ)
130		1nn 12176	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
131		ne0i 4269	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
132	130, 131	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ℕ ≠ ∅)
133	129, 132	eqnetrd 3001	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
134		dm0rn0 5866	. . . . . . . . . 10 ⊢ (dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ∅)
135	134	necon3bii 2986	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
136	133, 135	sylib 219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
137		ffn 6655	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) Fn ℕ)
138		breq1 5075	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) → (𝑧 ≤ 𝑥 ↔ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
139	138	ralrn 7029	. . . . . . . . . . 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 3163	. . . . . . . . 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 258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥)
143		supxrre 13270	. . . . . . . 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 1379	. . . . . . 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 25514	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
146	145	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → vol:dom vol⟶(0[,]+∞))
147	146, 17	cofmpt 7074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
148	147	rneqd 5880	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
149		rnco2 6205	. . . . . . . . 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 9349	. . . . . . 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 5100	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⇝ (vol‘(◡𝐹 “ {𝑘})))
155		i1ff 25661	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
156		frn 6662	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
157	3, 155, 156	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
158	157	ssdifssd 4077	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
159	158	sselda 3915	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ)
160	159	recnd 11164	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ)
161		nnex 12171	. . . . . 6 ⊢ ℕ ∈ V
162	161	mptex 7167	. . . . 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 7025	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ∈ ℝ)
165	164	recnd 11164	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ∈ ℂ)
166	54	oveq2d 7372	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
167		eqid 2739	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) = (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
168		ovex 7389	. . . . . . 7 ⊢ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))) ∈ V
169	166, 167, 168	fvmpt 6935	. . . . . 6 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
170	57	oveq2d 7372	. . . . . 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 482	. . . 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 15590	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) ⇝ (𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
174	161	mptex 7167	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ∈ V
175	174	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ∈ V)
176	159	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → 𝑘 ∈ ℝ)
177	176, 32	remulcld 11166	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ∈ ℝ)
178	177	fmpttd 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))):ℕ⟶ℝ)
179	178	ffvelcdmda 7025	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℝ)
180	179	recnd 11164	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℂ)
181	180	anasss 467	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (ran 𝐹 ∖ {0}) ∧ 𝑗 ∈ ℕ)) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℂ)
182	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ dom ∫1)
183		itg1climres.5	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
184	183	i1fres 25690	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫1 ∧ (𝐴‘𝑛) ∈ dom vol) → 𝐺 ∈ dom ∫1)
185	182, 14, 184	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ dom ∫1)
186	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ∈ Fin)
187		ffn 6655	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
188	3, 155, 187	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℝ)
189	188	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 Fn ℝ)
190		fnfvelrn 7021	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran 𝐹)
191	189, 190	sylan 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran 𝐹)
192		i1f0rn 25667	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹)
193	3, 192	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ran 𝐹)
194	193	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran 𝐹)
195	191, 194	ifcld 4501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ ran 𝐹)
196	195, 183	fmptd 7055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℝ⟶ran 𝐹)
197		frn 6662	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ran 𝐹 → ran 𝐺 ⊆ ran 𝐹)
198		ssdif 4074	. . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
201	200	ssdifd 4075	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))
202		itg1val2 25669	. . . . . . . 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 1380	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫1‘𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐺 “ {𝑘}))))
204		fvex 6840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘𝑥) ∈ V
205		c0ex 11129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
206	204, 205	ifex 4505	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ V
207	183	fvmpt2 6947	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ V) → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
208	206, 207	mpan2 697	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
209	208	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
210	209	eqeq1d 2741	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘))
211		eldifsni 4723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0)
212	211	ad2antlr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ≠ 0)
213		neeq1 2996	. . . . . . . . . . . . . . . . . . . 20 ⊢ (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ≠ 0 ↔ 𝑘 ≠ 0))
214	212, 213	syl5ibrcom 248	. . . . . . . . . . . . . . . . . . 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 4463	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 ∈ (𝐴‘𝑛) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 0)
216	215	necon1ai 2961	. . . . . . . . . . . . . . . . . . 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 567	. . . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . . 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 4460	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴‘𝑛) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = (𝐹‘𝑥))
221	220	eqeq1d 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴‘𝑛) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 ↔ (𝐹‘𝑥) = 𝑘))
222	221	pm5.32i 579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘) ↔ (𝑥 ∈ (𝐴‘𝑛) ∧ (𝐹‘𝑥) = 𝑘))
223	222	biancomi 463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘) ↔ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛)))
224	219, 223	bitrdi 288	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛))))
225	224	pm5.32da 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘) ↔ (𝑥 ∈ ℝ ∧ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛)))))
226		anass 469	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛)) ↔ (𝑥 ∈ ℝ ∧ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛))))
227	225, 226	bitr4di 290	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘) ↔ ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛))))
228		i1ff 25661	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
229		ffn 6655	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
230	185, 228, 229	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 Fn ℝ)
231	230	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐺 Fn ℝ)
232		fniniseg 7001	. . . . . . . . . . . . . 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 3899	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ (𝑥 ∈ (◡𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴‘𝑛)))
235	189	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ)
236		fniniseg 7001	. . . . . . . . . . . . . . . 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 637	. . . . . . . . . . . . . 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 284	. . . . . . . . . . . . 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 312	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
241	240	alrimiv 1934	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑥(𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
242		nfmpt1 5171	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
243	183, 242	nfcxfr 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐺
244	243	nfcnv 5820	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥◡𝐺
245		nfcv 2901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑘}
246	244, 245	nfima 6020	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(◡𝐺 “ {𝑘})
247		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))
248	246, 247	cleqf 2929	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑘}) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ ∀𝑥(𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
249	241, 248	sylibr 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐺 “ {𝑘}) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))
250	249	fveq2d 6831	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐺 “ {𝑘})) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
251	250	oveq2d 7372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐺 “ {𝑘}))) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
252	251	sumeq2dv 15655	. . . . . . 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 5165	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))))
255	254	fveq1d 6829	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (∫1‘𝐺))‘𝑗) = ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗))
256	166	sumeq2sdv 15656	. . . . . 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 2739	. . . . . 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 15641	. . . . . 6 ⊢ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))) ∈ V
259	256, 257, 258	fvmpt 6935	. . . . 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 15656	. . . . 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 15774	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
264		itg1val 25668	. . 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 5100	1 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ifcif 4454  {csn 4555  ∪ cuni 4838  ∪ ciun 4921   class class class wbr 5072   ↦ cmpt 5153  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   “ cima 5621   ∘ ccom 5622   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Fincfn 8883  supcsup 9343  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034  +∞cpnf 11167  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  ℕcn 12165  [,]cicc 13292   ⇝ cli 15437  Σcsu 15639  vol*covol 25447  volcvol 25448  ∫₁citg1 25600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-cc 10348  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-disj 5040  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ioo 13293  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-rlim 15442  df-sum 15640  df-rest 17376  df-topgen 17397  df-psmet 21339  df-xmet 21340  df-met 21341  df-bl 21342  df-mopn 21343  df-top 22877  df-topon 22894  df-bases 22929  df-cmp 23370  df-ovol 25449  df-vol 25450  df-mbf 25604  df-itg1 25605

This theorem is referenced by:  itg2monolem1  25735