Theorem itg2i1fseq 26081

Description: Subject to the conditions coming from mbfi1fseq 26047, 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 6885	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃‘𝑛) = (𝑃‘𝑚))
2	1	fveq1d 6887	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑚)‘𝑥))
3	2	cbvmptv 5208	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥))
4		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑃‘𝑚)‘𝑥) = ((𝑃‘𝑚)‘𝑦))
5	4	mpteq2dv 5198	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
6	3, 5	eqtrid 2807	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
7	6	rneqd 5919	. . . . 5 ⊢ (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)))
8	7	supeq1d 9441	. . . 4 ⊢ (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ))
9	8	cbvmptv 5208	. . 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 7084	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫1)
12		i1fmbf 26001	. . . 4 ⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚) ∈ MblFn)
13	11, 12	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ MblFn)
14		i1ff 26002	. . . . 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 5114	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (0𝑝 ∘r ≤ (𝑃‘𝑛) ↔ 0𝑝 ∘r ≤ (𝑃‘𝑚)))
18		fvoveq1 7443	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
19	1, 18	breq12d 5115	. . . . . . . 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 3575	. . . . . 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 25998	. . . 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 13590	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
28		itg2i1fseq.2	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
29	28	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞))
30	27, 29	sselid 3906	. . . 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 26080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘r ≤ 𝐹)
34	15	ffnd 6710	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) Fn ℝ)
35	28	ffnd 6710	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn ℝ)
36	35	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
37		reex 11294	. . . . . . . . . 10 ⊢ ℝ ∈ V
38	37	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
39		inidm 4123	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
40		eqidd 2761	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) = ((𝑃‘𝑚)‘𝑦))
41		eqidd 2761	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦))
42	34, 36, 38, 38, 39, 40, 41	ofrfval 7703	. . . . . . . 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 3254	. . . . . 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 3154	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))
47		brralrspcev 5164	. . . 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 6889	. . . . . 6 ⊢ (𝑛 = 𝑚 → (∫2‘(𝑃‘𝑛)) = (∫2‘(𝑃‘𝑚)))
50	49	cbvmptv 5208	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚)))
51	50	rneqi 5918	. . . 4 ⊢ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚)))
52	51	supeq1i 9442	. . 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 26079	. 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))), ℝ*, < ))
54	28	feqmptd 6953	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
55	1	fveq1d 6887	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘𝑚)‘𝑦))
56	55	cbvmptv 5208	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))
57	56	rneqi 5918	. . . . . . . 8 ⊢ ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))
58	57	supeq1i 9442	. . . . . . 7 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )
59		nnuz 13007	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
60		1zzd 12730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
61	15	ffvelcdmda 7084	. . . . . . . . . . 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 7114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)):ℕ⟶ℝ)
64		peano2nn 12350	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
65		ffvelcdm 7081	. . . . . . . . . . . . . . . . 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 26002	. . . . . . . . . . . . . . . 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 6710	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
70		eqidd 2761	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
71	34, 69, 38, 38, 39, 40, 70	ofrfval 7703	. . . . . . . . . . . . 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 3254	. . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))
76		fvex 6898	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑚)‘𝑦) ∈ V
77	55, 75, 76	fvmpt 6993	. . . . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 + 1) → (𝑃‘𝑛) = (𝑃‘(𝑚 + 1)))
80	79	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 + 1) → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
81		fvex 6898	. . . . . . . . . . . . 13 ⊢ ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
82	80, 75, 81	fvmpt 6993	. . . . . . . . . . . 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 5136	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)))
86	77	breq1d 5112	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧))
87	86	ralbiia 3106	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)
88	87	rexbii 3109	. . . . . . . . . 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 15842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ))
91		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑛)‘𝑦))
92	91	mpteq2dv 5198	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)))
93		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
94	92, 93	breq12d 5115	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)))
95	94	rspccva 3575	. . . . . . . . 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 15724	. . . . . . . 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 2809	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ) = (𝐹‘𝑦))
100	99	mpteq2dva 5197	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
101	54, 100	eqtr4d 2798	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )))
102	101, 9	eqtr4di 2813	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < )))
103	102	fveq2d 6889	. 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 26062	. . . . . . . 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 5197	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))))
108	104, 107	eqtr4id 2814	. . . . 5 ⊢ (𝜑 → 𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃‘𝑚))))
109	108, 50	eqtr4di 2813	. . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))))
110	109	rneqd 5919	. . 3 ⊢ (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))))
111	110	supeq1d 9441	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃‘𝑛))), ℝ*, < ))
112	53, 103, 111	3eqtr4d 2805	1 ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5650  ran crn 5651   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∘_r cofr 7692  supcsup 9435  ℝcr 11202  0cc0 11203  1c1 11204   + caddc 11206  +∞cpnf 11343  ℝ^*cxr 11345   < clt 11346   ≤ cle 11347  ℕcn 12338  [,)cico 13481   ⇝ cli 15656  MblFncmbf 25940  ∫₁citg1 25941  ∫₂citg2 25942  0_𝑝c0p 25995

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-cc 10516  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281  ax-addf 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-oadd 8483  df-omul 8484  df-er 8720  df-map 8852  df-pm 8853  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9507  df-dju 9985  df-card 10023  df-acn 10026  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-n0 12610  df-z 12697  df-uz 12969  df-q 13079  df-rp 13124  df-xneg 13244  df-xadd 13245  df-xmul 13246  df-ioo 13483  df-ioc 13484  df-ico 13485  df-icc 13486  df-fz 13643  df-fzo 13792  df-fl 13935  df-seq 14148  df-exp 14208  df-hash 14478  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-clim 15660  df-rlim 15661  df-sum 15859  df-rest 17598  df-topgen 17619  df-psmet 21675  df-xmet 21676  df-met 21677  df-bl 21678  df-mopn 21679  df-top 23217  df-topon 23234  df-bases 23269  df-cmp 23710  df-ovol 25790  df-vol 25791  df-mbf 25945  df-itg1 25946  df-itg2 25947  df-0p 25996

This theorem is used by:  itg2i1fseq2  26082