Theorem itg2cnlem1 25722

Description: Lemma for itgcn 25806. (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 6848	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ V
2		c0ex 11130	. . . . . . . . . 10 ⊢ 0 ∈ V
3	1, 2	ifex 4531	. . . . . . . . 9 ⊢ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ V
4		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
5	4	fvmpt2 6954	. . . . . . . . 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 692	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥) = if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
7	6	mpteq2dv 5193	. . . . . . 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 5888	. . . . . 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 9353	. . . . 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 5194	. . . 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 2899	. . . . 5 ⊢ Ⅎ𝑦sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)), ℝ, < )
12		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥ℕ
13		nfmpt1 5198	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
14	12, 13	nfmpt 5197	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
15		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚
16	14, 15	nffv 6845	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)
17		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
18	16, 17	nffv 6845	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)
19	12, 18	nfmpt 5197	. . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦))
20	19	nfrn 5902	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦))
21		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥ℝ
22		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥 <
23	20, 21, 22	nfsup 9358	. . . . 5 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < )
24		fveq2 6835	. . . . . . . . 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 5193	. . . . . . . 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 5103	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑥) ≤ 𝑛 ↔ (𝐹‘𝑥) ≤ 𝑚))
27	26	ifbid 4504	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
28	27	mpteq2dv 5193	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
29	28	fveq1d 6837	. . . . . . . . . 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 5203	. . . . . . . . 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 2737	. . . . . . . . . . . 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 11121	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
33	32	mptex 7171	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∈ V
34	28, 31, 33	fvmpt 6942	. . . . . . . . . . 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 6837	. . . . . . . . . 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 5194	. . . . . . . . 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 2763	. . . . . . . 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 2788	. . . . . . 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 5888	. . . . . 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 9353	. . . . 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 5201	. . . 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 2762	. . 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 6835	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
44	43	breq1d 5109	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚))
45	44, 43	ifbieq1d 4505	. . . . . 6 ⊢ (𝑥 = 𝑦 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
46	45	cbvmptv 5203	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
47	34	adantl 481	. . . . 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 12156	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
49	48	ad2antlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ)
50	49	rexrd 11186	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ*)
51		elioopnf 13363	. . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
54		itg2cn.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
55	54	ffnd 6664	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℝ)
56	55	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝐹 Fn ℝ)
57		elpreima 7005	. . . . . . . . . . . 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 708	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ (𝐹‘𝑦) ∈ (𝑚(,)+∞)))
60		rge0ssre 13376	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ ℝ
61		fss 6679	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
62	54, 60, 61	sylancl 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
63	62	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
64	63	ffvelcdmda 7031	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
65	64	biantrurd 532	. . . . . . . . . 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 3912	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ (𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
69	68	baib 535	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
70	69	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
71	64, 49	lenltd 11283	. . . . . . . 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 4504	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
74	73	mpteq2dva 5192	. . . . 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 2798	. . . 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 4089	. . . . . 6 ⊢ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ⊆ ℝ
77	76	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ⊆ ℝ)
78		rembl 25501	. . . . . 6 ⊢ ℝ ∈ dom vol
79	78	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ dom vol)
80		fvex 6848	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
81	80, 2	ifex 4531	. . . . . 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 4085	. . . . . . 7 ⊢ (𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) → ¬ 𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
84	83	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))) → ¬ 𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
85	84	iffalsed 4491	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = 0)
86		iftrue 4486	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = (𝐹‘𝑦))
87	86	mpteq2ia 5194	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) = (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ (𝐹‘𝑦))
88		resmpt 5997	. . . . . . . . 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 2763	. . . . . . 7 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) = ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
91	54	feqmptd 6903	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
92		itg2cn.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ MblFn)
93	91, 92	eqeltrrd 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn)
94		mbfima 25591	. . . . . . . . . 10 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
95	92, 62, 94	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
96		cmmbl 25495	. . . . . . . . 9 ⊢ ((◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
97	95, 96	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
98		mbfres 25605	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn ∧ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol) → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
99	93, 97, 98	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
100	90, 99	eqeltrid 2841	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
101	100	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
102	77, 79, 82, 85, 101	mbfss 25607	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
103	75, 102	eqeltrd 2837	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) ∈ MblFn)
104	54	ffvelcdmda 7031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
105		0e0icopnf 13378	. . . . . 6 ⊢ 0 ∈ (0[,)+∞)
106		ifcl 4526	. . . . . 6 ⊢ (((𝐹‘𝑥) ∈ (0[,)+∞) ∧ 0 ∈ (0[,)+∞)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
107	104, 105, 106	sylancl 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
108	107	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
109	47, 108	fmpt3d 7063	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚):ℝ⟶(0[,)+∞))
110		elrege0 13374	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
111	104, 110	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
112	111	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
113	112	adantlr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
114	113	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ∈ ℝ)
115	114	leidd 11707	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ (𝐹‘𝑥))
116		iftrue 4486	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ≤ 𝑚 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
117	116	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
118	48	ad3antlr 732	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → 𝑚 ∈ ℝ)
119		peano2re 11310	. . . . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ 𝑚)
122	118	lep1d 12077	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → 𝑚 ≤ (𝑚 + 1))
123	114, 118, 120, 121, 122	letrd 11294	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ (𝑚 + 1))
124	123	iftrued 4488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0) = (𝐹‘𝑥))
125	115, 117, 124	3brtr4d 5131	. . . . . . 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 4489	. . . . . . . . 9 ⊢ (¬ (𝐹‘𝑥) ≤ 𝑚 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = 0)
127	126	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = 0)
128	111	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐹‘𝑥))
129		0le0 12250	. . . . . . . . . . 11 ⊢ 0 ≤ 0
130		breq2 5103	. . . . . . . . . . . 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 5103	. . . . . . . . . . . 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 4520	. . . . . . . . . . 11 ⊢ ((0 ≤ (𝐹‘𝑥) ∧ 0 ≤ 0) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
133	128, 129, 132	sylancl 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
134	133	adantlr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
135	134	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹‘𝑥) ≤ 𝑚) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
136	127, 135	eqbrtrd 5121	. . . . . . 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 813	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
138	137	ralrimiva 3129	. . . . 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 4531	. . . . . . 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 2738	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
142		eqidd 2738	. . . . . 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 7645	. . . . 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 257	. . . 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 12161	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
146	145	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
147		breq2 5103	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘𝑥) ≤ 𝑛 ↔ (𝐹‘𝑥) ≤ (𝑚 + 1)))
148	147	ifbid 4504	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) = if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
149	148	mpteq2dv 5193	. . . . . 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 7171	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)) ∈ V
151	149, 31, 150	fvmpt 6942	. . . . 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 5131	. . 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	ffvelcdmda 7031	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
155	34	adantl 481	. . . . . . 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 6837	. . . . . 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 11707	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ≤ (𝐹‘𝑥))
158		breq1 5102	. . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 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 4520	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ≤ (𝐹‘𝑥) ∧ 0 ≤ (𝐹‘𝑥)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
161	157, 128, 160	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
162	161	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
163	162	ralrimiva 3129	. . . . . . . . . 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 4531	. . . . . . . . . . . 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 6903	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
168	167	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
169	164, 166, 113, 141, 168	ofrfval2 7645	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∘r ≤ 𝐹)
171	166	fmpttd 7062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)):ℝ⟶V)
172	171	ffnd 6664	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) Fn ℝ)
173	55	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
174		inidm 4180	. . . . . . . . . 10 ⊢ (ℝ ∩ ℝ) = ℝ
175		eqidd 2738	. . . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦))
177	172, 173, 164, 164, 174, 175, 176	ofrfval 7634	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
179	178	r19.21bi 3229	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
180	179	an32s 653	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
181	156, 180	eqbrtrd 5121	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
182	181	ralrimiva 3129	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
183		brralrspcev 5159	. . . 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 585	. . 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 6839	. . . . . . 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 5203	. . . . . 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 6839	. . . . . . 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 5194	. . . . . 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 2763	. . . . 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 5887	. . . 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 9354	. . 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 25714	. 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 2737	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
194	27, 193, 165	fvmpt 6942	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
195	194	adantl 481	. . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
197	195, 196	eqbrtrd 5121	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ≤ (𝐹‘𝑥))
198	197	ralrimiva 3129	. . . . . . . 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 7062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)):ℕ⟶V)
201	200	ffnd 6664	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ)
202		breq1 5102	. . . . . . . . . 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 7035	. . . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥))
206	112	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑥) ∈ ℝ)
207		0re 11138	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
208		ifcl 4526	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ ℝ)
209	206, 207, 208	sylancl 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ ℝ)
210	209	fmpttd 7062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)):ℕ⟶ℝ)
211	210	frnd 6671	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ⊆ ℝ)
212		1nn 12160	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
213	193, 209	dmmptd 6638	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ℕ)
214	212, 213	eleqtrrid 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
215		n0i 4293	. . . . . . . . . 10 ⊢ (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) → ¬ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅)
216		dm0rn0 5874	. . . . . . . . . . 11 ⊢ (dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅)
217	216	necon3bbii 2980	. . . . . . . . . 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 218	. . . . . . . . 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 5159	. . . . . . . . 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 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ 𝑧)
222		suprleub 12112	. . . . . . . 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 1376	. . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ≤ (𝐹‘𝑥))
225		arch 12402	. . . . . . . . 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 729	. . . . . . . . . 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 11225	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐹‘𝑥) < 𝑚 → (𝐹‘𝑥) ≤ 𝑚))
229	112, 48, 228	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑥) < 𝑚 → (𝐹‘𝑥) ≤ 𝑚))
230	229	impr 454	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → (𝐹‘𝑥) ≤ 𝑚)
231	230	iftrued 4488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
232	227, 231	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = (𝐹‘𝑥))
233	201	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ)
234		simprl 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → 𝑚 ∈ ℕ)
235		fnfvelrn 7027	. . . . . . . . . 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 585	. . . . . . . . 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 2838	. . . . . . . 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 3144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
239	211, 219, 221, 238	suprubd 12108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))
240	211, 219, 221	suprcld 12109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ∈ ℝ)
241	240, 112	letri3d 11279	. . . . . 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 714	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) = (𝐹‘𝑥))
243	242	mpteq2dva 5192	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < )) = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
244	243, 167	eqtr4d 2775	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < )) = 𝐹)
245	244	fveq2d 6839	. 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))) = (∫2‘𝐹))
246	192, 245	eqtr3d 2774	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4286  ifcif 4480   class class class wbr 5099   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ∘_r cofr 7623  supcsup 9347  ℝcr 11029  0cc0 11030  1c1 11031   + caddc 11033  +∞cpnf 11167  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  ℕcn 12149  (,)cioo 13265  [,)cico 13267  volcvol 25424  MblFncmbf 25575  ∫₂citg2 25577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cc 10349  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-disj 5067  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-ofr 7625  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-dju 9817  df-card 9855  df-acn 9858  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 12150  df-2 12212  df-3 12213  df-n0 12406  df-z 12493  df-uz 12756  df-q 12866  df-rp 12910  df-xneg 13030  df-xadd 13031  df-xmul 13032  df-ioo 13269  df-ioc 13270  df-ico 13271  df-icc 13272  df-fz 13428  df-fzo 13575  df-fl 13716  df-seq 13929  df-exp 13989  df-hash 14258  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-clim 15415  df-rlim 15416  df-sum 15614  df-rest 17346  df-topgen 17367  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-top 22842  df-topon 22859  df-bases 22894  df-cmp 23335  df-ovol 25425  df-vol 25426  df-mbf 25580  df-itg1 25581  df-itg2 25582

This theorem is referenced by:  itg2cn  25724