Theorem itg2i1fseq 24825

Description: Subject to the conditions coming from mbfi1fseq 24791, 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 6756	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃‘𝑛) = (𝑃‘𝑚))
2	1	fveq1d 6758	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑚)‘𝑥))
3	2	cbvmptv 5183	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥))
4		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑃‘𝑚)‘𝑥) = ((𝑃‘𝑚)‘𝑦))
5	4	mpteq2dv 5172	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
6	3, 5	syl5eq 2791	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
7	6	rneqd 5836	. . . . 5 ⊢ (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
8	7	supeq1d 9135	. . . 4 ⊢ (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ))
9	8	cbvmptv 5183	. . 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	ffvelrnda 6943	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫1)
12		i1fmbf 24744	. . . 4 ⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚) ∈ MblFn)
13	11, 12	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ MblFn)
14		i1ff 24745	. . . . 5 ⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚):ℝ⟶ℝ)
15	11, 14	syl 17	. . . 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 5082	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (0𝑝 ∘r ≤ (𝑃‘𝑛) ↔ 0𝑝 ∘r ≤ (𝑃‘𝑚)))
18		fvoveq1 7278	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
19	1, 18	breq12d 5083	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
20	17, 19	anbi12d 630	. . . . . . 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 3551	. . . . . 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 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (0𝑝 ∘r ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
23	22	simpld 494	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0𝑝 ∘r ≤ (𝑃‘𝑚))
24		0plef 24741	. . . 4 ⊢ ((𝑃‘𝑚):ℝ⟶(0[,)+∞) ↔ ((𝑃‘𝑚):ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ (𝑃‘𝑚)))
25	15, 23, 24	sylanbrc 582	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚):ℝ⟶(0[,)+∞))
26	22	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))
27		rge0ssre 13117	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
28		itg2i1fseq.2	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
29	28	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞))
30	27, 29	sselid 3915	. . . 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 24824	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘r ≤ 𝐹)
34	15	ffnd 6585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) Fn ℝ)
35	28	ffnd 6585	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn ℝ)
36	35	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
37		reex 10893	. . . . . . . . . 10 ⊢ ℝ ∈ V
38	37	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
39		inidm 4149	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
40		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) = ((𝑃‘𝑚)‘𝑦))
41		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦))
42	34, 36, 38, 38, 39, 40, 41	ofrfval 7521	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)))
43	33, 42	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
44	43	r19.21bi 3132	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
45	44	an32s 648	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
46	45	ralrimiva 3107	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
47		brralrspcev 5130	. . . 4 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
48	30, 46, 47	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
49	1	fveq2d 6760	. . . . . 6 ⊢ (𝑛 = 𝑚 → (∫2‘(𝑃‘𝑛)) = (∫2‘(𝑃‘𝑚)))
50	49	cbvmptv 5183	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚)))
51	50	rneqi 5835	. . . 4 ⊢ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚)))
52	51	supeq1i 9136	. . 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 24823	. 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))), ℝ*, < ))
54	28	feqmptd 6819	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
55	1	fveq1d 6758	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘𝑚)‘𝑦))
56	55	cbvmptv 5183	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))
57	56	rneqi 5835	. . . . . . . 8 ⊢ ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))
58	57	supeq1i 9136	. . . . . . 7 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )
59		nnuz 12550	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
60		1zzd 12281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
61	15	ffvelrnda 6943	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ)
62	61	an32s 648	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ)
63	62, 56	fmptd 6970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)):ℕ⟶ℝ)
64		peano2nn 11915	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
65		ffvelrn 6941	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:ℕ⟶dom ∫1 ∧ (𝑚 + 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
66	10, 64, 65	syl2an 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
67		i1ff 24745	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
68	66, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
69	68	ffnd 6585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
70		eqidd 2739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
71	34, 69, 38, 38, 39, 40, 70	ofrfval 7521	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)))
72	26, 71	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
73	72	r19.21bi 3132	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
74	73	an32s 648	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
75		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))
76		fvex 6769	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑚)‘𝑦) ∈ V
77	55, 75, 76	fvmpt 6857	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦))
78	77	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦))
79		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 + 1) → (𝑃‘𝑛) = (𝑃‘(𝑚 + 1)))
80	79	fveq1d 6758	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 + 1) → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
81		fvex 6769	. . . . . . . . . . . . 13 ⊢ ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
82	80, 75, 81	fvmpt 6857	. . . . . . . . . . . 12 ⊢ ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
83	64, 82	syl 17	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
84	83	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
85	74, 78, 84	3brtr4d 5102	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)))
86	77	breq1d 5080	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧))
87	86	ralbiia 3089	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
88	87	rexbii 3177	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
89	48, 88	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧)
90	59, 60, 63, 85, 89	climsup 15309	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ))
91		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑛)‘𝑦))
92	91	mpteq2dv 5172	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)))
93		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
94	92, 93	breq12d 5083	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)))
95	94	rspccva 3551	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))
96	32, 95	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))
97		climuni 15189	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = (𝐹‘𝑦))
99	58, 98	eqtr3id 2793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ) = (𝐹‘𝑦))
100	99	mpteq2dva 5170	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
101	54, 100	eqtr4d 2781	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )))
102	101, 9	eqtr4di 2797	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < )))
103	102	fveq2d 6760	. 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 24806	. . . . . . . 8 ⊢ (((𝑃‘𝑚) ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ (𝑃‘𝑚)) → (∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚)))
106	11, 23, 105	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚)))
107	106	mpteq2dva 5170	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))))
108	104, 107	eqtr4id 2798	. . . . 5 ⊢ (𝜑 → 𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚))))
109	108, 50	eqtr4di 2797	. . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))))
110	109	rneqd 5836	. . 3 ⊢ (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))))
111	110	supeq1d 9135	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))), ℝ*, < ))
112	53, 103, 111	3eqtr4d 2788	1 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_r cofr 7510  supcsup 9129  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941  ℕcn 11903  [,)cico 13010   ⇝ cli 15121  MblFncmbf 24683  ∫₁citg1 24684  ∫₂citg2 24685  0_𝑝c0p 24738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cc 10122  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-acn 9631  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ioc 13013  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-rest 17050  df-topgen 17071  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-top 21951  df-topon 21968  df-bases 22004  df-cmp 22446  df-ovol 24533  df-vol 24534  df-mbf 24688  df-itg1 24689  df-itg2 24690  df-0p 24739

This theorem is referenced by:  itg2i1fseq2  24826