Theorem itg2i1fseq 26109

Description: Subject to the conditions coming from mbfi1fseq 26075, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)

Hypotheses

Ref	Expression
itg2i1fseq.1	⊢ (𝜑 → 𝐹 ∈ MblFn)
itg2i1fseq.2	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
itg2i1fseq.3	⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)
itg2i1fseq.4	⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
itg2i1fseq.5	⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
itg2i1fseq.6	⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚)))

Assertion

Ref	Expression
itg2i1fseq	⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < ))

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

Allowed substitution hints:   𝜑(𝑥, 𝑛)   𝑆(𝑥, 𝑚, 𝑛)

Proof of Theorem itg2i1fseq

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

Step	Hyp	Ref	Expression
1		fveq2 6911	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃‘𝑛) = (𝑃‘𝑚))
2	1	fveq1d 6913	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑚)‘𝑥))
3	2	cbvmptv 5213	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥))
4		fveq2 6911	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑃‘𝑚)‘𝑥) = ((𝑃‘𝑚)‘𝑦))
5	4	mpteq2dv 5203	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
6	3, 5	eqtrid 2810	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
7	6	rneqd 5928	. . . . 5 ⊢ (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
8	7	supeq1d 9468	. . . 4 ⊢ (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ))
9	8	cbvmptv 5213	. . 3 ⊢ (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ))
10		itg2i1fseq.3	. . . . 5 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)
11	10	ffvelcdmda 7111	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫1)
12		i1fmbf 26029	. . . 4 ⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚) ∈ MblFn)
13	11, 12	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ MblFn)
14		i1ff 26030	. . . . 5 ⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚):ℝ⟶ℝ)
15	11, 14	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚):ℝ⟶ℝ)
16		itg2i1fseq.4	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
17	1	breq2d 5118	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (0𝑝 ∘r ≤ (𝑃‘𝑛) ↔ 0𝑝 ∘r ≤ (𝑃‘𝑚)))
18		fvoveq1 7469	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
19	1, 18	breq12d 5119	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
20	17, 19	anbi12d 644	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝 ∘r ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))))
21	20	rspccva 3579	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ∧ 𝑚 ∈ ℕ) → (0𝑝 ∘r ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
22	16, 21	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (0𝑝 ∘r ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
23	22	simpld 500	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0𝑝 ∘r ≤ (𝑃‘𝑚))
24		0plef 26026	. . . 4 ⊢ ((𝑃‘𝑚):ℝ⟶(0[,)+∞) ↔ ((𝑃‘𝑚):ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ (𝑃‘𝑚)))
25	15, 23, 24	sylanbrc 595	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚):ℝ⟶(0[,)+∞))
26	22	simprd 501	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))
27		rge0ssre 13617	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
28		itg2i1fseq.2	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
29	28	ffvelcdmda 7111	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞))
30	27, 29	sselid 3911	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
31		itg2i1fseq.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ MblFn)
32		itg2i1fseq.5	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
33	31, 28, 10, 16, 32	itg2i1fseqle 26108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘r ≤ 𝐹)
34	15	ffnd 6735	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) Fn ℝ)
35	28	ffnd 6735	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn ℝ)
36	35	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
37		reex 11321	. . . . . . . . . 10 ⊢ ℝ ∈ V
38	37	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
39		inidm 4126	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
40		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) = ((𝑃‘𝑚)‘𝑦))
41		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦))
42	34, 36, 38, 38, 39, 40, 41	ofrfval 7730	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)))
43	33, 42	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
44	43	r19.21bi 3258	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
45	44	an32s 665	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
46	45	ralrimiva 3158	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
47		brralrspcev 5169	. . . 4 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
48	30, 46, 47	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
49	1	fveq2d 6915	. . . . . 6 ⊢ (𝑛 = 𝑚 → (∫2‘(𝑃‘𝑛)) = (∫2‘(𝑃‘𝑚)))
50	49	cbvmptv 5213	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚)))
51	50	rneqi 5927	. . . 4 ⊢ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚)))
52	51	supeq1i 9469	. . 3 ⊢ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚))), ℝ*, < )
53	9, 13, 25, 26, 48, 52	itg2mono 26107	. 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))), ℝ*, < ))
54	28	feqmptd 6979	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
55	1	fveq1d 6913	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘𝑚)‘𝑦))
56	55	cbvmptv 5213	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))
57	56	rneqi 5927	. . . . . . . 8 ⊢ ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))
58	57	supeq1i 9469	. . . . . . 7 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )
59		nnuz 13034	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
60		1zzd 12757	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
61	15	ffvelcdmda 7111	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ)
62	61	an32s 665	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ)
63	62, 56	fmptd 7141	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)):ℕ⟶ℝ)
64		peano2nn 12377	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
65		ffvelcdm 7108	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:ℕ⟶dom ∫1 ∧ (𝑚 + 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
66	10, 64, 65	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
67		i1ff 26030	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
68	66, 67	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
69	68	ffnd 6735	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
70		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
71	34, 69, 38, 38, 39, 40, 70	ofrfval 7730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)))
72	26, 71	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
73	72	r19.21bi 3258	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
74	73	an32s 665	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
75		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))
76		fvex 6924	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑚)‘𝑦) ∈ V
77	55, 75, 76	fvmpt 7019	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦))
78	77	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦))
79		fveq2 6911	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 + 1) → (𝑃‘𝑛) = (𝑃‘(𝑚 + 1)))
80	79	fveq1d 6913	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 + 1) → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
81		fvex 6924	. . . . . . . . . . . . 13 ⊢ ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
82	80, 75, 81	fvmpt 7019	. . . . . . . . . . . 12 ⊢ ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
83	64, 82	syl 18	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
84	83	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
85	74, 78, 84	3brtr4d 5141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)))
86	77	breq1d 5116	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧))
87	86	ralbiia 3110	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
88	87	rexbii 3113	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
89	48, 88	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧)
90	59, 60, 63, 85, 89	climsup 15869	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ))
91		fveq2 6911	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑛)‘𝑦))
92	91	mpteq2dv 5203	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)))
93		fveq2 6911	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
94	92, 93	breq12d 5119	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)))
95	94	rspccva 3579	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))
96	32, 95	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))
97		climuni 15751	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) ∧ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = (𝐹‘𝑦))
98	90, 96, 97	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = (𝐹‘𝑦))
99	58, 98	eqtr3id 2812	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ) = (𝐹‘𝑦))
100	99	mpteq2dva 5202	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
101	54, 100	eqtr4d 2801	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )))
102	101, 9	eqtr4di 2816	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < )))
103	102	fveq2d 6915	. 2 ⊢ (𝜑 → (∫2‘𝐹) = (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))))
104		itg2i1fseq.6	. . . . . 6 ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚)))
105		itg2itg1 26090	. . . . . . . 8 ⊢ (((𝑃‘𝑚) ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ (𝑃‘𝑚)) → (∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚)))
106	11, 23, 105	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚)))
107	106	mpteq2dva 5202	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))))
108	104, 107	eqtr4id 2817	. . . . 5 ⊢ (𝜑 → 𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚))))
109	108, 50	eqtr4di 2816	. . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))))
110	109	rneqd 5928	. . 3 ⊢ (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))))
111	110	supeq1d 9468	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))), ℝ*, < ))
112	53, 103, 111	3eqtr4d 2808	1 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   class class class wbr 5106   ↦ cmpt 5190  dom cdm 5656  ran crn 5657   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   ∘_r cofr 7719  supcsup 9462  ℝcr 11229  0cc0 11230  1c1 11231   + caddc 11233  +∞cpnf 11370  ℝ^*cxr 11372   < clt 11373   ≤ cle 11374  ℕcn 12365  [,)cico 13508   ⇝ cli 15683  MblFncmbf 25968  ∫₁citg1 25969  ∫₂citg2 25970  0_𝑝c0p 26023

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-cc 10543  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308  ax-addf 11309

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-disj 5074  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-of 7720  df-ofr 7721  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-er 8748  df-map 8880  df-pm 8881  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-fi 9433  df-sup 9464  df-inf 9465  df-oi 9534  df-dju 10012  df-card 10050  df-acn 10053  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-n0 12637  df-z 12724  df-uz 12996  df-q 13106  df-rp 13151  df-xneg 13271  df-xadd 13272  df-xmul 13273  df-ioo 13510  df-ioc 13511  df-ico 13512  df-icc 13513  df-fz 13670  df-fzo 13819  df-fl 13962  df-seq 14175  df-exp 14235  df-hash 14505  df-cj 15298  df-re 15299  df-im 15300  df-sqrt 15434  df-abs 15435  df-clim 15687  df-rlim 15688  df-sum 15886  df-rest 17625  df-topgen 17646  df-psmet 21703  df-xmet 21704  df-met 21705  df-bl 21706  df-mopn 21707  df-top 23245  df-topon 23262  df-bases 23297  df-cmp 23738  df-ovol 25818  df-vol 25819  df-mbf 25973  df-itg1 25974  df-itg2 25975  df-0p 26024

This theorem is used by:  itg2i1fseq2  26110