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Theorem itg2i1fseq 25120
Description: Subject to the conditions coming from mbfi1fseq 25086, 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 6842 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃𝑛) = (𝑃𝑚))
21fveq1d 6844 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑚)‘𝑥))
32cbvmptv 5218 . . . . . . 7 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥))
4 fveq2 6842 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑚)‘𝑥) = ((𝑃𝑚)‘𝑦))
54mpteq2dv 5207 . . . . . . 7 (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
63, 5eqtrid 2788 . . . . . 6 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
76rneqd 5893 . . . . 5 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
87supeq1d 9382 . . . 4 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ))
98cbvmptv 5218 . . 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)
1110ffvelcdmda 7035 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ dom ∫1)
12 i1fmbf 25039 . . . 4 ((𝑃𝑚) ∈ dom ∫1 → (𝑃𝑚) ∈ MblFn)
1311, 12syl 17 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ MblFn)
14 i1ff 25040 . . . . 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 5117 . . . . . . . 8 (𝑛 = 𝑚 → (0𝑝r ≤ (𝑃𝑛) ↔ 0𝑝r ≤ (𝑃𝑚)))
18 fvoveq1 7380 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
191, 18breq12d 5118 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2017, 19anbi12d 631 . . . . . . 7 (𝑛 = 𝑚 → ((0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))))
2120rspccva 3580 . . . . . 6 ((∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ∧ 𝑚 ∈ ℕ) → (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2216, 21sylan 580 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2322simpld 495 . . . 4 ((𝜑𝑚 ∈ ℕ) → 0𝑝r ≤ (𝑃𝑚))
24 0plef 25036 . . . 4 ((𝑃𝑚):ℝ⟶(0[,)+∞) ↔ ((𝑃𝑚):ℝ⟶ℝ ∧ 0𝑝r ≤ (𝑃𝑚)))
2515, 23, 24sylanbrc 583 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚):ℝ⟶(0[,)+∞))
2622simprd 496 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))
27 rge0ssre 13373 . . . . 5 (0[,)+∞) ⊆ ℝ
28 itg2i1fseq.2 . . . . . 6 (𝜑𝐹:ℝ⟶(0[,)+∞))
2928ffvelcdmda 7035 . . . . 5 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
3027, 29sselid 3942 . . . 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 25119 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘r𝐹)
3415ffnd 6669 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) Fn ℝ)
3528ffnd 6669 . . . . . . . . . 10 (𝜑𝐹 Fn ℝ)
3635adantr 481 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
37 reex 11142 . . . . . . . . . 10 ℝ ∈ V
3837a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
39 inidm 4178 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
40 eqidd 2737 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) = ((𝑃𝑚)‘𝑦))
41 eqidd 2737 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
4234, 36, 38, 38, 39, 40, 41ofrfval 7627 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘r𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)))
4333, 42mpbid 231 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4443r19.21bi 3234 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4544an32s 650 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4645ralrimiva 3143 . . . 4 ((𝜑𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
47 brralrspcev 5165 . . . 4 (((𝐹𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
4830, 46, 47syl2anc 584 . . 3 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
491fveq2d 6846 . . . . . 6 (𝑛 = 𝑚 → (∫2‘(𝑃𝑛)) = (∫2‘(𝑃𝑚)))
5049cbvmptv 5218 . . . . 5 (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5150rneqi 5892 . . . 4 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5251supeq1i 9383 . . 3 sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))), ℝ*, < )
539, 13, 25, 26, 48, 52itg2mono 25118 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
5428feqmptd 6910 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
551fveq1d 6844 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑚)‘𝑦))
5655cbvmptv 5218 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5756rneqi 5892 . . . . . . . 8 ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5857supeq1i 9383 . . . . . . 7 sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )
59 nnuz 12806 . . . . . . . . 9 ℕ = (ℤ‘1)
60 1zzd 12534 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 1 ∈ ℤ)
6115ffvelcdmda 7035 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6261an32s 650 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6362, 56fmptd 7062 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)):ℕ⟶ℝ)
64 peano2nn 12165 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
65 ffvelcdm 7032 . . . . . . . . . . . . . . . . 17 ((𝑃:ℕ⟶dom ∫1 ∧ (𝑚 + 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
6610, 64, 65syl2an 596 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
67 i1ff 25040 . . . . . . . . . . . . . . . 16 ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6866, 67syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6968ffnd 6669 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
70 eqidd 2737 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
7134, 69, 38, 38, 39, 40, 70ofrfval 7627 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)))
7226, 71mpbid 231 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7372r19.21bi 3234 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7473an32s 650 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
75 eqid 2736 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))
76 fvex 6855 . . . . . . . . . . . 12 ((𝑃𝑚)‘𝑦) ∈ V
7755, 75, 76fvmpt 6948 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
7877adantl 482 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
79 fveq2 6842 . . . . . . . . . . . . . 14 (𝑛 = (𝑚 + 1) → (𝑃𝑛) = (𝑃‘(𝑚 + 1)))
8079fveq1d 6844 . . . . . . . . . . . . 13 (𝑛 = (𝑚 + 1) → ((𝑃𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
81 fvex 6855 . . . . . . . . . . . . 13 ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
8280, 75, 81fvmpt 6948 . . . . . . . . . . . 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 482 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8574, 78, 843brtr4d 5137 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)))
8677breq1d 5115 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃𝑚)‘𝑦) ≤ 𝑧))
8786ralbiia 3094 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8887rexbii 3097 . . . . . . . . . 10 (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8948, 88sylibr 233 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧)
9059, 60, 63, 85, 89climsup 15554 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ))
91 fveq2 6842 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑛)‘𝑦))
9291mpteq2dv 5207 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)))
93 fveq2 6842 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
9492, 93breq12d 5118 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)))
9594rspccva 3580 . . . . . . . . 9 ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
9632, 95sylan 580 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
97 climuni 15434 . . . . . . . 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 584 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = (𝐹𝑦))
9958, 98eqtr3id 2790 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ) = (𝐹𝑦))
10099mpteq2dva 5205 . . . . 5 (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
10154, 100eqtr4d 2779 . . . 4 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )))
102101, 9eqtr4di 2794 . . 3 (𝜑𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < )))
103102fveq2d 6846 . 2 (𝜑 → (∫2𝐹) = (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))))
104 itg2i1fseq.6 . . . . . 6 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))
105 itg2itg1 25101 . . . . . . . 8 (((𝑃𝑚) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑃𝑚)) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
10611, 23, 105syl2anc 584 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
107106mpteq2dva 5205 . . . . . 6 (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚))))
108104, 107eqtr4id 2795 . . . . 5 (𝜑𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))))
109108, 50eqtr4di 2794 . . . 4 (𝜑𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
110109rneqd 5893 . . 3 (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
111110supeq1d 9382 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
11253, 103, 1113eqtr4d 2786 1 (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  r cofr 7616  supcsup 9376  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cn 12153  [,)cico 13266  cli 15366  MblFncmbf 24978  1citg1 24979  2citg2 24980  0𝑝c0p 25033
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984  df-itg2 24985  df-0p 25034
This theorem is referenced by:  itg2i1fseq2  25121
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