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Theorem itg2i1fseq 24357
 Description: Subject to the conditions coming from mbfi1fseq 24323, 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 (𝜑𝐹 ∈ MblFn)
itg2i1fseq.2 (𝜑𝐹:ℝ⟶(0[,)+∞))
itg2i1fseq.3 (𝜑𝑃:ℕ⟶dom ∫1)
itg2i1fseq.4 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
itg2i1fseq.5 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
itg2i1fseq.6 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))
Assertion
Ref Expression
itg2i1fseq (𝜑 → (∫2𝐹) = 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.
StepHypRef Expression
1 fveq2 6652 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃𝑛) = (𝑃𝑚))
21fveq1d 6654 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑚)‘𝑥))
32cbvmptv 5145 . . . . . . 7 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥))
4 fveq2 6652 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑚)‘𝑥) = ((𝑃𝑚)‘𝑦))
54mpteq2dv 5138 . . . . . . 7 (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
63, 5syl5eq 2869 . . . . . 6 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
76rneqd 5785 . . . . 5 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
87supeq1d 8898 . . . 4 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ))
98cbvmptv 5145 . . 3 (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ))
10 itg2i1fseq.3 . . . . 5 (𝜑𝑃:ℕ⟶dom ∫1)
1110ffvelrnda 6833 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ dom ∫1)
12 i1fmbf 24277 . . . 4 ((𝑃𝑚) ∈ dom ∫1 → (𝑃𝑚) ∈ MblFn)
1311, 12syl 17 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ MblFn)
14 i1ff 24278 . . . . 5 ((𝑃𝑚) ∈ dom ∫1 → (𝑃𝑚):ℝ⟶ℝ)
1511, 14syl 17 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚):ℝ⟶ℝ)
16 itg2i1fseq.4 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
171breq2d 5054 . . . . . . . 8 (𝑛 = 𝑚 → (0𝑝r ≤ (𝑃𝑛) ↔ 0𝑝r ≤ (𝑃𝑚)))
18 fvoveq1 7163 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
191, 18breq12d 5055 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2017, 19anbi12d 633 . . . . . . 7 (𝑛 = 𝑚 → ((0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))))
2120rspccva 3597 . . . . . 6 ((∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ∧ 𝑚 ∈ ℕ) → (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2216, 21sylan 583 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2322simpld 498 . . . 4 ((𝜑𝑚 ∈ ℕ) → 0𝑝r ≤ (𝑃𝑚))
24 0plef 24274 . . . 4 ((𝑃𝑚):ℝ⟶(0[,)+∞) ↔ ((𝑃𝑚):ℝ⟶ℝ ∧ 0𝑝r ≤ (𝑃𝑚)))
2515, 23, 24sylanbrc 586 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚):ℝ⟶(0[,)+∞))
2622simprd 499 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))
27 rge0ssre 12834 . . . . 5 (0[,)+∞) ⊆ ℝ
28 itg2i1fseq.2 . . . . . 6 (𝜑𝐹:ℝ⟶(0[,)+∞))
2928ffvelrnda 6833 . . . . 5 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
3027, 29sseldi 3940 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
31 itg2i1fseq.1 . . . . . . . . 9 (𝜑𝐹 ∈ MblFn)
32 itg2i1fseq.5 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
3331, 28, 10, 16, 32itg2i1fseqle 24356 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘r𝐹)
3415ffnd 6495 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) Fn ℝ)
3528ffnd 6495 . . . . . . . . . 10 (𝜑𝐹 Fn ℝ)
3635adantr 484 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
37 reex 10617 . . . . . . . . . 10 ℝ ∈ V
3837a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
39 inidm 4169 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
40 eqidd 2823 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) = ((𝑃𝑚)‘𝑦))
41 eqidd 2823 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
4234, 36, 38, 38, 39, 40, 41ofrfval 7402 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘r𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)))
4333, 42mpbid 235 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4443r19.21bi 3198 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4544an32s 651 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4645ralrimiva 3174 . . . 4 ((𝜑𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
47 brralrspcev 5102 . . . 4 (((𝐹𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
4830, 46, 47syl2anc 587 . . 3 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
491fveq2d 6656 . . . . . 6 (𝑛 = 𝑚 → (∫2‘(𝑃𝑛)) = (∫2‘(𝑃𝑚)))
5049cbvmptv 5145 . . . . 5 (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5150rneqi 5784 . . . 4 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5251supeq1i 8899 . . 3 sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))), ℝ*, < )
539, 13, 25, 26, 48, 52itg2mono 24355 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
5428feqmptd 6715 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
551fveq1d 6654 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑚)‘𝑦))
5655cbvmptv 5145 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5756rneqi 5784 . . . . . . . 8 ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5857supeq1i 8899 . . . . . . 7 sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )
59 nnuz 12269 . . . . . . . . 9 ℕ = (ℤ‘1)
60 1zzd 12001 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 1 ∈ ℤ)
6115ffvelrnda 6833 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6261an32s 651 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6362, 56fmptd 6860 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)):ℕ⟶ℝ)
64 peano2nn 11637 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
65 ffvelrn 6831 . . . . . . . . . . . . . . . . 17 ((𝑃:ℕ⟶dom ∫1 ∧ (𝑚 + 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
6610, 64, 65syl2an 598 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
67 i1ff 24278 . . . . . . . . . . . . . . . 16 ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6866, 67syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6968ffnd 6495 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
70 eqidd 2823 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
7134, 69, 38, 38, 39, 40, 70ofrfval 7402 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)))
7226, 71mpbid 235 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7372r19.21bi 3198 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7473an32s 651 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
75 eqid 2822 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))
76 fvex 6665 . . . . . . . . . . . 12 ((𝑃𝑚)‘𝑦) ∈ V
7755, 75, 76fvmpt 6750 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
7877adantl 485 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
79 fveq2 6652 . . . . . . . . . . . . . 14 (𝑛 = (𝑚 + 1) → (𝑃𝑛) = (𝑃‘(𝑚 + 1)))
8079fveq1d 6654 . . . . . . . . . . . . 13 (𝑛 = (𝑚 + 1) → ((𝑃𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
81 fvex 6665 . . . . . . . . . . . . 13 ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
8280, 75, 81fvmpt 6750 . . . . . . . . . . . 12 ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8364, 82syl 17 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8483adantl 485 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8574, 78, 843brtr4d 5074 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)))
8677breq1d 5052 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃𝑚)‘𝑦) ≤ 𝑧))
8786ralbiia 3156 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8887rexbii 3235 . . . . . . . . . 10 (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8948, 88sylibr 237 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧)
9059, 60, 63, 85, 89climsup 15017 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ))
91 fveq2 6652 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑛)‘𝑦))
9291mpteq2dv 5138 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)))
93 fveq2 6652 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
9492, 93breq12d 5055 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)))
9594rspccva 3597 . . . . . . . . 9 ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
9632, 95sylan 583 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
97 climuni 14900 . . . . . . . 8 (((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) ∧ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = (𝐹𝑦))
9890, 96, 97syl2anc 587 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = (𝐹𝑦))
9958, 98syl5eqr 2871 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ) = (𝐹𝑦))
10099mpteq2dva 5137 . . . . 5 (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
10154, 100eqtr4d 2860 . . . 4 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )))
102101, 9eqtr4di 2875 . . 3 (𝜑𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < )))
103102fveq2d 6656 . 2 (𝜑 → (∫2𝐹) = (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))))
104 itg2itg1 24338 . . . . . . . 8 (((𝑃𝑚) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑃𝑚)) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
10511, 23, 104syl2anc 587 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
106105mpteq2dva 5137 . . . . . 6 (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚))))
107 itg2i1fseq.6 . . . . . 6 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))
108106, 107syl6reqr 2876 . . . . 5 (𝜑𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))))
109108, 50eqtr4di 2875 . . . 4 (𝜑𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
110109rneqd 5785 . . 3 (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
111110supeq1d 8898 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
11253, 103, 1113eqtr4d 2867 1 (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131  Vcvv 3469   class class class wbr 5042   ↦ cmpt 5122  dom cdm 5532  ran crn 5533   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140   ∘r cofr 7393  supcsup 8892  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529  +∞cpnf 10661  ℝ*cxr 10663   < clt 10664   ≤ cle 10665  ℕcn 11625  [,)cico 12728   ⇝ cli 14832  MblFncmbf 24216  ∫1citg1 24217  ∫2citg2 24218  0𝑝c0p 24271 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-cc 9846  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-disj 5008  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-ofr 7395  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836  df-rlim 14837  df-sum 15034  df-rest 16687  df-topgen 16708  df-psmet 20081  df-xmet 20082  df-met 20083  df-bl 20084  df-mopn 20085  df-top 21497  df-topon 21514  df-bases 21549  df-cmp 21990  df-ovol 24066  df-vol 24067  df-mbf 24221  df-itg1 24222  df-itg2 24223  df-0p 24272 This theorem is referenced by:  itg2i1fseq2  24358
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