Theorem itg2cnlem1 24924

Description: Lemma for itgcn 25007. (Contributed by Mario Carneiro, 30-Aug-2014.)

Hypotheses

Ref	Expression
itg2cn.1	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
itg2cn.2	⊢ (𝜑 → 𝐹 ∈ MblFn)
itg2cn.3	⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)

Assertion

Ref	Expression
itg2cnlem1	⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))), ℝ*, < ) = (∫2‘𝐹))

Distinct variable groups:   𝑥,𝑛,𝐹   𝜑,𝑛,𝑥

Proof of Theorem itg2cnlem1

Dummy variables 𝑚 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6784	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ V
2		c0ex 10970	. . . . . . . . . 10 ⊢ 0 ∈ V
3	1, 2	ifex 4515	. . . . . . . . 9 ⊢ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ V
4		eqid 2740	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
5	4	fvmpt2 6883	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥) = if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
6	3, 5	mpan2 688	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥) = if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
7	6	mpteq2dv 5181	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)) = (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
8	7	rneqd 5846	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)) = ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
9	8	supeq1d 9183	. . . . 5 ⊢ (𝑥 ∈ ℝ → sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))
10	9	mpteq2ia 5182	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)), ℝ, < )) = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))
11		nfcv 2909	. . . . 5 ⊢ Ⅎ𝑦sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)), ℝ, < )
12		nfcv 2909	. . . . . . . 8 ⊢ Ⅎ𝑥ℕ
13		nfmpt1 5187	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
14	12, 13	nfmpt 5186	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
15		nfcv 2909	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚
16	14, 15	nffv 6781	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)
17		nfcv 2909	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
18	16, 17	nffv 6781	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)
19	12, 18	nfmpt 5186	. . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦))
20	19	nfrn 5860	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦))
21		nfcv 2909	. . . . . 6 ⊢ Ⅎ𝑥ℝ
22		nfcv 2909	. . . . . 6 ⊢ Ⅎ𝑥 <
23	20, 21, 22	nfsup 9188	. . . . 5 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < )
24		fveq2 6771	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑦))
25	24	mpteq2dv 5181	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑦)))
26		breq2 5083	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑥) ≤ 𝑛 ↔ (𝐹‘𝑥) ≤ 𝑚))
27	26	ifbid 4488	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
28	27	mpteq2dv 5181	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
29	28	fveq1d 6773	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦))
30	29	cbvmptv 5192	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦))
31		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))) = (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
32		reex 10963	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
33	32	mptex 7096	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∈ V
34	28, 31, 33	fvmpt 6872	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
35	34	fveq1d 6773	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦))
36	35	mpteq2ia 5182	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦))
37	30, 36	eqtr4i 2771	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑦)) = (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦))
38	25, 37	eqtrdi 2796	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)) = (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)))
39	38	rneqd 5846	. . . . . 6 ⊢ (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)) = ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)))
40	39	supeq1d 9183	. . . . 5 ⊢ (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < ))
41	11, 23, 40	cbvmpt 5190	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < ))
42	10, 41	eqtr3i 2770	. . 3 ⊢ (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < ))
43		fveq2 6771	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
44	43	breq1d 5089	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚))
45	44, 43	ifbieq1d 4489	. . . . . 6 ⊢ (𝑥 = 𝑦 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
46	45	cbvmptv 5192	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
47	34	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
48		nnre 11980	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
49	48	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ)
50	49	rexrd 11026	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ*)
51		elioopnf 13174	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℝ* → ((𝐹‘𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 < (𝐹‘𝑦))))
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 < (𝐹‘𝑦))))
53		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
54		itg2cn.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
55	54	ffnd 6599	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℝ)
56	55	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝐹 Fn ℝ)
57		elpreima 6932	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℝ → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ (𝑚(,)+∞))))
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ (𝑚(,)+∞))))
59	53, 58	mpbirand 704	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ (𝐹‘𝑦) ∈ (𝑚(,)+∞)))
60		rge0ssre 13187	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ ℝ
61		fss 6615	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
62	54, 60, 61	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
63	62	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
64	63	ffvelrnda 6958	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
65	64	biantrurd 533	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑚 < (𝐹‘𝑦) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 < (𝐹‘𝑦))))
66	52, 59, 65	3bitr4d 311	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ 𝑚 < (𝐹‘𝑦)))
67	66	notbid 318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ ¬ 𝑚 < (𝐹‘𝑦)))
68		eldif 3902	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ (𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
69	68	baib 536	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
70	69	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
71	64, 49	lenltd 11121	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑦) ≤ 𝑚 ↔ ¬ 𝑚 < (𝐹‘𝑦)))
72	67, 70, 71	3bitr4d 311	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ (𝐹‘𝑦) ≤ 𝑚))
73	72	ifbid 4488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
74	73	mpteq2dva 5179	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0)))
75	46, 47, 74	3eqtr4a 2806	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) = (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)))
76		difss 4071	. . . . . 6 ⊢ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ⊆ ℝ
77	76	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ⊆ ℝ)
78		rembl 24702	. . . . . 6 ⊢ ℝ ∈ dom vol
79	78	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ dom vol)
80		fvex 6784	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
81	80, 2	ifex 4515	. . . . . 6 ⊢ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) ∈ V
82	81	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) ∈ V)
83		eldifn 4067	. . . . . . 7 ⊢ (𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) → ¬ 𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
84	83	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))) → ¬ 𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
85	84	iffalsed 4476	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = 0)
86		iftrue 4471	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = (𝐹‘𝑦))
87	86	mpteq2ia 5182	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) = (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ (𝐹‘𝑦))
88		resmpt 5944	. . . . . . . . 9 ⊢ ((ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) = (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ (𝐹‘𝑦)))
89	76, 88	ax-mp 5	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) = (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ (𝐹‘𝑦))
90	87, 89	eqtr4i 2771	. . . . . . 7 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) = ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
91	54	feqmptd 6834	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
92		itg2cn.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ MblFn)
93	91, 92	eqeltrrd 2842	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn)
94		mbfima 24792	. . . . . . . . . 10 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
95	92, 62, 94	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
96		cmmbl 24696	. . . . . . . . 9 ⊢ ((◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
97	95, 96	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
98		mbfres 24806	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn ∧ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol) → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
99	93, 97, 98	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
100	90, 99	eqeltrid 2845	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
101	100	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
102	77, 79, 82, 85, 101	mbfss 24808	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
103	75, 102	eqeltrd 2841	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) ∈ MblFn)
104	54	ffvelrnda 6958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
105		0e0icopnf 13189	. . . . . 6 ⊢ 0 ∈ (0[,)+∞)
106		ifcl 4510	. . . . . 6 ⊢ (((𝐹‘𝑥) ∈ (0[,)+∞) ∧ 0 ∈ (0[,)+∞)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
107	104, 105, 106	sylancl 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
108	107	adantlr 712	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
109	47, 108	fmpt3d 6987	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚):ℝ⟶(0[,)+∞))
110		elrege0 13185	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
111	104, 110	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
112	111	simpld 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
113	112	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
114	113	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ∈ ℝ)
115	114	leidd 11541	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ (𝐹‘𝑥))
116		iftrue 4471	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ≤ 𝑚 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
117	116	adantl 482	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
118	48	ad3antlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → 𝑚 ∈ ℝ)
119		peano2re 11148	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
120	118, 119	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝑚 + 1) ∈ ℝ)
121		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ 𝑚)
122	118	lep1d 11906	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → 𝑚 ≤ (𝑚 + 1))
123	114, 118, 120, 121, 122	letrd 11132	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ (𝑚 + 1))
124	123	iftrued 4473	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0) = (𝐹‘𝑥))
125	115, 117, 124	3brtr4d 5111	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
126		iffalse 4474	. . . . . . . . 9 ⊢ (¬ (𝐹‘𝑥) ≤ 𝑚 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = 0)
127	126	adantl 482	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = 0)
128	111	simprd 496	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐹‘𝑥))
129		0le0 12074	. . . . . . . . . . 11 ⊢ 0 ≤ 0
130		breq2 5083	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) = if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0) → (0 ≤ (𝐹‘𝑥) ↔ 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
131		breq2 5083	. . . . . . . . . . . 12 ⊢ (0 = if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0) → (0 ≤ 0 ↔ 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
132	130, 131	ifboth 4504	. . . . . . . . . . 11 ⊢ ((0 ≤ (𝐹‘𝑥) ∧ 0 ≤ 0) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
133	128, 129, 132	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
134	133	adantlr 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
135	134	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹‘𝑥) ≤ 𝑚) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
136	127, 135	eqbrtrd 5101	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
137	125, 136	pm2.61dan 810	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
138	137	ralrimiva 3110	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
139	1, 2	ifex 4515	. . . . . . 7 ⊢ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0) ∈ V
140	139	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0) ∈ V)
141		eqidd 2741	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
142		eqidd 2741	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
143	79, 108, 140, 141, 142	ofrfval2 7548	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)) ↔ ∀𝑥 ∈ ℝ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
144	138, 143	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
145		peano2nn 11985	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
146	145	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
147		breq2 5083	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘𝑥) ≤ 𝑛 ↔ (𝐹‘𝑥) ≤ (𝑚 + 1)))
148	147	ifbid 4488	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) = if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
149	148	mpteq2dv 5181	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
150	32	mptex 7096	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)) ∈ V
151	149, 31, 150	fvmpt 6872	. . . . 5 ⊢ ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘(𝑚 + 1)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
152	146, 151	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘(𝑚 + 1)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)))
153	144, 47, 152	3brtr4d 5111	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) ∘r ≤ ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘(𝑚 + 1)))
154	62	ffvelrnda 6958	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
155	34	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
156	155	fveq1d 6773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦))
157	112	leidd 11541	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ≤ (𝐹‘𝑥))
158		breq1 5082	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) → ((𝐹‘𝑥) ≤ (𝐹‘𝑥) ↔ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥)))
159		breq1 5082	. . . . . . . . . . . . . 14 ⊢ (0 = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) → (0 ≤ (𝐹‘𝑥) ↔ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥)))
160	158, 159	ifboth 4504	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ≤ (𝐹‘𝑥) ∧ 0 ≤ (𝐹‘𝑥)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
161	157, 128, 160	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
162	161	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
163	162	ralrimiva 3110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
164	32	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
165	1, 2	ifex 4515	. . . . . . . . . . . 12 ⊢ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ V
166	165	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ V)
167	54	feqmptd 6834	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
168	167	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
169	164, 166, 113, 141, 168	ofrfval2 7548	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥)))
170	163, 169	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∘r ≤ 𝐹)
171	166	fmpttd 6986	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)):ℝ⟶V)
172	171	ffnd 6599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) Fn ℝ)
173	55	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
174		inidm 4158	. . . . . . . . . 10 ⊢ (ℝ ∩ ℝ) = ℝ
175		eqidd 2741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦))
176		eqidd 2741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦))
177	172, 173, 164, 164, 174, 175, 176	ofrfval 7537	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦)))
178	170, 177	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
179	178	r19.21bi 3135	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
180	179	an32s 649	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
181	156, 180	eqbrtrd 5101	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
182	181	ralrimiva 3110	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
183		brralrspcev 5139	. . . 4 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ 𝑧)
184	154, 182, 183	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ 𝑧)
185	28	fveq2d 6775	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))))
186	185	cbvmptv 5192	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))))
187	34	fveq2d 6775	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))))
188	187	mpteq2ia 5182	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))))
189	186, 188	eqtr4i 2771	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))) = (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)))
190	189	rneqi 5845	. . . 4 ⊢ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))) = ran (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)))
191	190	supeq1i 9184	. . 3 ⊢ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚))), ℝ*, < )
192	42, 103, 109, 153, 184, 191	itg2mono 24916	. 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))), ℝ*, < ))
193		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
194	27, 193, 165	fvmpt 6872	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
195	194	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
196	161	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
197	195, 196	eqbrtrd 5101	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ≤ (𝐹‘𝑥))
198	197	ralrimiva 3110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ≤ (𝐹‘𝑥))
199	3	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ V)
200	199	fmpttd 6986	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)):ℕ⟶V)
201	200	ffnd 6599	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ)
202		breq1 5082	. . . . . . . . . 10 ⊢ (𝑤 = ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) → (𝑤 ≤ (𝐹‘𝑥) ↔ ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ≤ (𝐹‘𝑥)))
203	202	ralrn 6961	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ → (∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ≤ (𝐹‘𝑥)))
204	201, 203	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ≤ (𝐹‘𝑥)))
205	198, 204	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥))
206	112	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑥) ∈ ℝ)
207		0re 10978	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
208		ifcl 4510	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ ℝ)
209	206, 207, 208	sylancl 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ ℝ)
210	209	fmpttd 6986	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)):ℕ⟶ℝ)
211	210	frnd 6606	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ⊆ ℝ)
212		1nn 11984	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
213	193, 209	dmmptd 6576	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ℕ)
214	212, 213	eleqtrrid 2848	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
215		n0i 4273	. . . . . . . . . 10 ⊢ (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) → ¬ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅)
216		dm0rn0 5833	. . . . . . . . . . 11 ⊢ (dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅)
217	216	necon3bbii 2993	. . . . . . . . . 10 ⊢ (¬ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ≠ ∅)
218	215, 217	sylib 217	. . . . . . . . 9 ⊢ (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) → ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ≠ ∅)
219	214, 218	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ≠ ∅)
220		brralrspcev 5139	. . . . . . . . 9 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ 𝑧)
221	112, 205, 220	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ 𝑧)
222		suprleub 11941	. . . . . . . 8 ⊢ (((ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ 𝑧) ∧ (𝐹‘𝑥) ∈ ℝ) → (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ≤ (𝐹‘𝑥) ↔ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥)))
223	211, 219, 221, 112, 222	syl31anc 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ≤ (𝐹‘𝑥) ↔ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥)))
224	205, 223	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ≤ (𝐹‘𝑥))
225		arch 12230	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ ℝ → ∃𝑚 ∈ ℕ (𝐹‘𝑥) < 𝑚)
226	112, 225	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑚 ∈ ℕ (𝐹‘𝑥) < 𝑚)
227	194	ad2antrl 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
228		ltle 11064	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐹‘𝑥) < 𝑚 → (𝐹‘𝑥) ≤ 𝑚))
229	112, 48, 228	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑥) < 𝑚 → (𝐹‘𝑥) ≤ 𝑚))
230	229	impr 455	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → (𝐹‘𝑥) ≤ 𝑚)
231	230	iftrued 4473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
232	227, 231	eqtrd 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = (𝐹‘𝑥))
233	201	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ)
234		simprl 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → 𝑚 ∈ ℕ)
235		fnfvelrn 6955	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
236	233, 234, 235	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
237	232, 236	eqeltrrd 2842	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → (𝐹‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
238	226, 237	rexlimddv 3222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
239	211, 219, 221, 238	suprubd 11937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))
240	211, 219, 221	suprcld 11938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ∈ ℝ)
241	240, 112	letri3d 11117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) = (𝐹‘𝑥) ↔ (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ≤ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))))
242	224, 239, 241	mpbir2and 710	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) = (𝐹‘𝑥))
243	242	mpteq2dva 5179	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < )) = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
244	243, 167	eqtr4d 2783	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < )) = 𝐹)
245	244	fveq2d 6775	. 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))) = (∫2‘𝐹))
246	192, 245	eqtr3d 2782	1 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))), ℝ*, < ) = (∫2‘𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ≠ wne 2945  ∀wral 3066  ∃wrex 3067  Vcvv 3431   ∖ cdif 3889   ⊆ wss 3892  ∅c0 4262  ifcif 4465   class class class wbr 5079   ↦ cmpt 5162  ^◡ccnv 5589  dom cdm 5590  ran crn 5591   ↾ cres 5592   “ cima 5593   Fn wfn 6427  ⟶wf 6428  ‘cfv 6432  (class class class)co 7271   ∘_r cofr 7526  supcsup 9177  ℝcr 10871  0cc0 10872  1c1 10873   + caddc 10875  +∞cpnf 11007  ℝ^*cxr 11009   < clt 11010   ≤ cle 11011  ℕcn 11973  (,)cioo 13078  [,)cico 13080  volcvol 24625  MblFncmbf 24776  ∫₂citg2 24778

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377  ax-cc 10192  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949  ax-pre-sup 10950

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-disj 5045  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-of 7527  df-ofr 7528  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-2o 8289  df-oadd 8292  df-omul 8293  df-er 8481  df-map 8600  df-pm 8601  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-fi 9148  df-sup 9179  df-inf 9180  df-oi 9247  df-dju 9660  df-card 9698  df-acn 9701  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12582  df-q 12688  df-rp 12730  df-xneg 12847  df-xadd 12848  df-xmul 12849  df-ioo 13082  df-ioc 13083  df-ico 13084  df-icc 13085  df-fz 13239  df-fzo 13382  df-fl 13510  df-seq 13720  df-exp 13781  df-hash 14043  df-cj 14808  df-re 14809  df-im 14810  df-sqrt 14944  df-abs 14945  df-clim 15195  df-rlim 15196  df-sum 15396  df-rest 17131  df-topgen 17152  df-psmet 20587  df-xmet 20588  df-met 20589  df-bl 20590  df-mopn 20591  df-top 22041  df-topon 22058  df-bases 22094  df-cmp 22536  df-ovol 24626  df-vol 24627  df-mbf 24781  df-itg1 24782  df-itg2 24783

This theorem is referenced by:  itg2cn  24926