Theorem itg2cnlem1 25899

Description: Lemma for itgcn 25983. (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 6894	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ V
2		c0ex 11199	. . . . . . . . . 10 ⊢ 0 ∈ V
3	1, 2	ifex 4537	. . . . . . . . 9 ⊢ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ V
4		eqid 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
5	4	fvmpt2 7001	. . . . . . . . 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 703	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥) = if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
7	6	mpteq2dv 5204	. . . . . . 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 5928	. . . . . 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 9405	. . . . 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 5205	. . . 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 2923	. . . . 5 ⊢ Ⅎ𝑦sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑥)), ℝ, < )
12		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥ℕ
13		nfmpt1 5209	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
14	12, 13	nfmpt 5208	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
15		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚
16	14, 15	nffv 6891	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)
17		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
18	16, 17	nffv 6891	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)
19	12, 18	nfmpt 5208	. . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦))
20	19	nfrn 5942	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦))
21		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥ℝ
22		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥 <
23	20, 21, 22	nfsup 9410	. . . . 5 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < )
24		fveq2 6881	. . . . . . . . 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 5204	. . . . . . . 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 5112	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑥) ≤ 𝑛 ↔ (𝐹‘𝑥) ≤ 𝑚))
27	26	ifbid 4510	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
28	27	mpteq2dv 5204	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
29	28	fveq1d 6883	. . . . . . . . . 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 5214	. . . . . . . . 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 2761	. . . . . . . . . . . 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 11190	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
33	32	mptex 7221	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∈ V
34	28, 31, 33	fvmpt 6989	. . . . . . . . . . 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 6883	. . . . . . . . . 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 5205	. . . . . . . . 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 2787	. . . . . . . 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 2812	. . . . . . 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 5928	. . . . . 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 9405	. . . . 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 5212	. . . 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 2786	. . 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 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
44	43	breq1d 5118	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚))
45	44, 43	ifbieq1d 4511	. . . . . 6 ⊢ (𝑥 = 𝑦 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
46	45	cbvmptv 5214	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
47	34	adantl 486	. . . . 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 12239	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
49	48	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ)
50	49	rexrd 11258	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ*)
51		elioopnf 13469	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℝ* → ((𝐹‘𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 < (𝐹‘𝑦))))
52	50, 51	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 < (𝐹‘𝑦))))
53		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
54		itg2cn.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
55	54	ffnd 6706	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℝ)
56	55	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝐹 Fn ℝ)
57		elpreima 7053	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℝ → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ (𝑚(,)+∞))))
58	56, 57	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ (𝑚(,)+∞))))
59	53, 58	mpbirand 719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ (𝐹‘𝑦) ∈ (𝑚(,)+∞)))
60		rge0ssre 13482	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ ℝ
61		fss 6722	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
62	54, 60, 61	sylancl 597	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
63	62	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
64	63	ffvelcdmda 7079	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
65	64	biantrurd 541	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑚 < (𝐹‘𝑦) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 < (𝐹‘𝑦))))
66	52, 59, 65	3bitr4d 314	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ 𝑚 < (𝐹‘𝑦)))
67	66	notbid 321	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞)) ↔ ¬ 𝑚 < (𝐹‘𝑦)))
68		eldif 3914	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ (𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
69	68	baib 544	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
70	69	adantl 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (◡𝐹 “ (𝑚(,)+∞))))
71	64, 49	lenltd 11355	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑦) ≤ 𝑚 ↔ ¬ 𝑚 < (𝐹‘𝑦)))
72	67, 70, 71	3bitr4d 314	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↔ (𝐹‘𝑦) ≤ 𝑚))
73	72	ifbid 4510	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = if((𝐹‘𝑦) ≤ 𝑚, (𝐹‘𝑦), 0))
74	73	mpteq2dva 5203	. . . . 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 2822	. . . 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 25678	. . . . . 6 ⊢ ℝ ∈ dom vol
79	78	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ dom vol)
80		fvex 6894	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
81	80, 2	ifex 4537	. . . . . 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 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))) → ¬ 𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
85	84	iffalsed 4497	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = 0)
86		iftrue 4492	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) → if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0) = (𝐹‘𝑦))
87	86	mpteq2ia 5205	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) = (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ (𝐹‘𝑦))
88		resmpt 6039	. . . . . . . . 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 2787	. . . . . . 7 ⊢ (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) = ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))))
91	54	feqmptd 6949	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
92		itg2cn.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ MblFn)
93	91, 92	eqeltrrd 2862	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn)
94		mbfima 25768	. . . . . . . . . 10 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
95	92, 62, 94	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
96		cmmbl 25672	. . . . . . . . 9 ⊢ ((◡𝐹 “ (𝑚(,)+∞)) ∈ dom vol → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
97	95, 96	syl 18	. . . . . . . 8 ⊢ (𝜑 → (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
98		mbfres 25782	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn ∧ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ∈ dom vol) → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
99	93, 97, 98	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
100	90, 99	eqeltrid 2865	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
101	100	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
102	77, 79, 82, 85, 101	mbfss 25784	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (◡𝐹 “ (𝑚(,)+∞))), (𝐹‘𝑦), 0)) ∈ MblFn)
103	75, 102	eqeltrd 2861	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚) ∈ MblFn)
104	54	ffvelcdmda 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
105		0e0icopnf 13484	. . . . . 6 ⊢ 0 ∈ (0[,)+∞)
106		ifcl 4532	. . . . . 6 ⊢ (((𝐹‘𝑥) ∈ (0[,)+∞) ∧ 0 ∈ (0[,)+∞)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
107	104, 105, 106	sylancl 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
108	107	adantlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ∈ (0[,)+∞))
109	47, 108	fmpt3d 7111	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚):ℝ⟶(0[,)+∞))
110		elrege0 13480	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
111	104, 110	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
112	111	simpld 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
113	112	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
114	113	adantr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ∈ ℝ)
115	114	leidd 11779	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ (𝐹‘𝑥))
116		iftrue 4492	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ≤ 𝑚 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
117	116	adantl 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
118	48	ad3antlr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → 𝑚 ∈ ℝ)
119		peano2re 11382	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
120	118, 119	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝑚 + 1) ∈ ℝ)
121		simpr 489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ 𝑚)
122	118	lep1d 12145	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → 𝑚 ≤ (𝑚 + 1))
123	114, 118, 120, 121, 122	letrd 11366	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → (𝐹‘𝑥) ≤ (𝑚 + 1))
124	123	iftrued 4494	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0) = (𝐹‘𝑥))
125	115, 117, 124	3brtr4d 5142	. . . . . . 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 4495	. . . . . . . . 9 ⊢ (¬ (𝐹‘𝑥) ≤ 𝑚 → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = 0)
127	126	adantl 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹‘𝑥) ≤ 𝑚) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = 0)
128	111	simprd 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐹‘𝑥))
129		0le0 12341	. . . . . . . . . . 11 ⊢ 0 ≤ 0
130		breq2 5112	. . . . . . . . . . . 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 5112	. . . . . . . . . . . 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 4526	. . . . . . . . . . 11 ⊢ ((0 ≤ (𝐹‘𝑥) ∧ 0 ≤ 0) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
133	128, 129, 132	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
134	133	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
135	134	adantr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹‘𝑥) ≤ 𝑚) → 0 ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
136	127, 135	eqbrtrd 5132	. . . . . . 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 824	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
138	137	ralrimiva 3155	. . . . 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 4537	. . . . . . 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 2762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)))
142		eqidd 2762	. . . . . 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 7695	. . . . 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 260	. . . 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 12244	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
146	145	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
147		breq2 5112	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘𝑥) ≤ 𝑛 ↔ (𝐹‘𝑥) ≤ (𝑚 + 1)))
148	147	ifbid 4510	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) = if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0))
149	148	mpteq2dv 5204	. . . . . 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 7221	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ (𝑚 + 1), (𝐹‘𝑥), 0)) ∈ V
151	149, 31, 150	fvmpt 6989	. . . . 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 18	. . . 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 5142	. . 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 7079	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
155	34	adantl 486	. . . . . . 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 6883	. . . . . 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 11779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ≤ (𝐹‘𝑥))
158		breq1 5111	. . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . 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 4526	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ≤ (𝐹‘𝑥) ∧ 0 ≤ (𝐹‘𝑥)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
161	157, 128, 160	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
162	161	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
163	162	ralrimiva 3155	. . . . . . . . . 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 4537	. . . . . . . . . . . 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 6949	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
168	167	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
169	164, 166, 113, 141, 168	ofrfval2 7695	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) ∘r ≤ 𝐹)
171	166	fmpttd 7110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)):ℝ⟶V)
172	171	ffnd 6706	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0)) Fn ℝ)
173	55	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
174		inidm 4178	. . . . . . . . . 10 ⊢ (ℝ ∩ ℝ) = ℝ
175		eqidd 2762	. . . . . . . . . 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 2762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦))
177	172, 173, 164, 164, 174, 175, 176	ofrfval 7684	. . . . . . . . 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 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
179	178	r19.21bi 3255	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
180	179	an32s 664	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))‘𝑦) ≤ (𝐹‘𝑦))
181	156, 180	eqbrtrd 5132	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
182	181	ralrimiva 3155	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
183		brralrspcev 5170	. . . 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 595	. . 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 6885	. . . . . . 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 5214	. . . . . 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 6885	. . . . . . 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 5205	. . . . . 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 2787	. . . . 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 5927	. . . 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 9406	. . 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 25891	. 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 2761	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))
194	27, 193, 165	fvmpt 6989	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0))
195	194	adantl 486	. . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) ≤ (𝐹‘𝑥))
197	195, 196	eqbrtrd 5132	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) ≤ (𝐹‘𝑥))
198	197	ralrimiva 3155	. . . . . . . 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 7110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)):ℕ⟶V)
201	200	ffnd 6706	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ)
202		breq1 5111	. . . . . . . . . 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 7083	. . . . . . . . 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 18	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ (𝐹‘𝑥))
206	112	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑥) ∈ ℝ)
207		0re 11209	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
208		ifcl 4532	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ ℝ)
209	206, 207, 208	sylancl 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0) ∈ ℝ)
210	209	fmpttd 7110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)):ℕ⟶ℝ)
211	210	frnd 6714	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ⊆ ℝ)
212		1nn 12243	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
213	193, 209	dmmptd 6680	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ℕ)
214	212, 213	eleqtrrid 2868	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
215		n0i 4292	. . . . . . . . . 10 ⊢ (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) → ¬ dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅)
216		dm0rn0 5914	. . . . . . . . . . 11 ⊢ (dom (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) = ∅)
217	216	necon3bbii 3003	. . . . . . . . . 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 221	. . . . . . . . 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 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) ≠ ∅)
220		brralrspcev 5170	. . . . . . . . 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 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))𝑤 ≤ 𝑧)
222		suprleub 12180	. . . . . . . 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 1398	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ≤ (𝐹‘𝑥))
225		arch 12500	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ ℝ → ∃𝑚 ∈ ℕ (𝐹‘𝑥) < 𝑚)
226	112, 225	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑚 ∈ ℕ (𝐹‘𝑥) < 𝑚)
227	194	ad2antrl 740	. . . . . . . . . 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 11297	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐹‘𝑥) < 𝑚 → (𝐹‘𝑥) ≤ 𝑚))
229	112, 48, 228	syl2an 607	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑥) < 𝑚 → (𝐹‘𝑥) ≤ 𝑚))
230	229	impr 459	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → (𝐹‘𝑥) ≤ 𝑚)
231	230	iftrued 4494	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → if((𝐹‘𝑥) ≤ 𝑚, (𝐹‘𝑥), 0) = (𝐹‘𝑥))
232	227, 231	eqtrd 2796	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0))‘𝑚) = (𝐹‘𝑥))
233	201	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)) Fn ℕ)
234		simprl 782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹‘𝑥) < 𝑚)) → 𝑚 ∈ ℕ)
235		fnfvelrn 7075	. . . . . . . . . 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 595	. . . . . . . . 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 2862	. . . . . . . 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 3170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))
239	211, 219, 221, 238	suprubd 12176	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))
240	211, 219, 221	suprcld 12177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) ∈ ℝ)
241	240, 112	letri3d 11351	. . . . . 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 725	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ) = (𝐹‘𝑥))
243	242	mpteq2dva 5203	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < )) = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
244	243, 167	eqtr4d 2799	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < )) = 𝐹)
245	244	fveq2d 6885	. 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)), ℝ, < ))) = (∫2‘𝐹))
246	192, 245	eqtr3d 2798	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 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3453   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  ifcif 4486   class class class wbr 5108   ↦ cmpt 5191  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∘_r cofr 7673  supcsup 9399  ℝcr 11098  0cc0 11099  1c1 11100   + caddc 11102  +∞cpnf 11239  ℝ^*cxr 11241   < clt 11242   ≤ cle 11243  ℕcn 12232  (,)cioo 13371  [,)cico 13373  volcvol 25601  MblFncmbf 25752  ∫₂citg2 25754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cc 10418  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-disj 5076  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-omul 8457  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-dju 9886  df-card 9924  df-acn 9927  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ioo 13375  df-ioc 13376  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13683  df-fl 13825  df-seq 14038  df-exp 14098  df-hash 14367  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-clim 15539  df-rlim 15540  df-sum 15738  df-rest 17474  df-topgen 17495  df-psmet 21493  df-xmet 21494  df-met 21495  df-bl 21496  df-mopn 21497  df-top 23030  df-topon 23047  df-bases 23082  df-cmp 23523  df-ovol 25602  df-vol 25603  df-mbf 25757  df-itg1 25758  df-itg2 25759

This theorem is referenced by:  itg2cn  25901