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Theorem itg2i1fseq 23970
 Description: Subject to the conditions coming from mbfi1fseq 23936, 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 6448 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃𝑛) = (𝑃𝑚))
21fveq1d 6450 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑚)‘𝑥))
32cbvmptv 4987 . . . . . . 7 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥))
4 fveq2 6448 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑚)‘𝑥) = ((𝑃𝑚)‘𝑦))
54mpteq2dv 4982 . . . . . . 7 (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
63, 5syl5eq 2826 . . . . . 6 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
76rneqd 5600 . . . . 5 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
87supeq1d 8642 . . . 4 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ))
98cbvmptv 4987 . . 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 6625 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ dom ∫1)
12 i1fmbf 23890 . . . 4 ((𝑃𝑚) ∈ dom ∫1 → (𝑃𝑚) ∈ MblFn)
1311, 12syl 17 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ MblFn)
14 i1ff 23891 . . . . 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 4900 . . . . . . . 8 (𝑛 = 𝑚 → (0𝑝𝑟 ≤ (𝑃𝑛) ↔ 0𝑝𝑟 ≤ (𝑃𝑚)))
18 fvoveq1 6947 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
191, 18breq12d 4901 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))))
2017, 19anbi12d 624 . . . . . . 7 (𝑛 = 𝑚 → ((0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝𝑟 ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)))))
2120rspccva 3510 . . . . . 6 ((∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) ∧ 𝑚 ∈ ℕ) → (0𝑝𝑟 ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))))
2216, 21sylan 575 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (0𝑝𝑟 ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))))
2322simpld 490 . . . 4 ((𝜑𝑚 ∈ ℕ) → 0𝑝𝑟 ≤ (𝑃𝑚))
24 0plef 23887 . . . 4 ((𝑃𝑚):ℝ⟶(0[,)+∞) ↔ ((𝑃𝑚):ℝ⟶ℝ ∧ 0𝑝𝑟 ≤ (𝑃𝑚)))
2515, 23, 24sylanbrc 578 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚):ℝ⟶(0[,)+∞))
2622simprd 491 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)))
27 rge0ssre 12599 . . . . 5 (0[,)+∞) ⊆ ℝ
28 itg2i1fseq.2 . . . . . 6 (𝜑𝐹:ℝ⟶(0[,)+∞))
2928ffvelrnda 6625 . . . . 5 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
3027, 29sseldi 3819 . . . 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 23969 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘𝑟𝐹)
3415ffnd 6294 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) Fn ℝ)
3528ffnd 6294 . . . . . . . . . 10 (𝜑𝐹 Fn ℝ)
3635adantr 474 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
37 reex 10365 . . . . . . . . . 10 ℝ ∈ V
3837a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
39 inidm 4043 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
40 eqidd 2779 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) = ((𝑃𝑚)‘𝑦))
41 eqidd 2779 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
4234, 36, 38, 38, 39, 40, 41ofrfval 7184 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘𝑟𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)))
4333, 42mpbid 224 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4443r19.21bi 3114 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4544an32s 642 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4645ralrimiva 3148 . . . 4 ((𝜑𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
47 brralrspcev 4948 . . . 4 (((𝐹𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
4830, 46, 47syl2anc 579 . . 3 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
491fveq2d 6452 . . . . . 6 (𝑛 = 𝑚 → (∫2‘(𝑃𝑛)) = (∫2‘(𝑃𝑚)))
5049cbvmptv 4987 . . . . 5 (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5150rneqi 5599 . . . 4 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5251supeq1i 8643 . . 3 sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))), ℝ*, < )
539, 13, 25, 26, 48, 52itg2mono 23968 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
5428feqmptd 6511 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
551fveq1d 6450 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑚)‘𝑦))
5655cbvmptv 4987 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5756rneqi 5599 . . . . . . . 8 ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5857supeq1i 8643 . . . . . . 7 sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )
59 nnuz 12034 . . . . . . . . 9 ℕ = (ℤ‘1)
60 1zzd 11765 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 1 ∈ ℤ)
6115ffvelrnda 6625 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6261an32s 642 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6362, 56fmptd 6650 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)):ℕ⟶ℝ)
64 peano2nn 11393 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
65 ffvelrn 6623 . . . . . . . . . . . . . . . . 17 ((𝑃:ℕ⟶dom ∫1 ∧ (𝑚 + 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
6610, 64, 65syl2an 589 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
67 i1ff 23891 . . . . . . . . . . . . . . . 16 ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6866, 67syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6968ffnd 6294 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
70 eqidd 2779 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
7134, 69, 38, 38, 39, 40, 70ofrfval 7184 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)))
7226, 71mpbid 224 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7372r19.21bi 3114 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7473an32s 642 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
75 eqid 2778 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))
76 fvex 6461 . . . . . . . . . . . 12 ((𝑃𝑚)‘𝑦) ∈ V
7755, 75, 76fvmpt 6544 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
7877adantl 475 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
79 fveq2 6448 . . . . . . . . . . . . . 14 (𝑛 = (𝑚 + 1) → (𝑃𝑛) = (𝑃‘(𝑚 + 1)))
8079fveq1d 6450 . . . . . . . . . . . . 13 (𝑛 = (𝑚 + 1) → ((𝑃𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
81 fvex 6461 . . . . . . . . . . . . 13 ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
8280, 75, 81fvmpt 6544 . . . . . . . . . . . 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 475 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8574, 78, 843brtr4d 4920 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)))
8677breq1d 4898 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃𝑚)‘𝑦) ≤ 𝑧))
8786ralbiia 3161 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8887rexbii 3224 . . . . . . . . . 10 (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8948, 88sylibr 226 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧)
9059, 60, 63, 85, 89climsup 14817 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ))
91 fveq2 6448 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑛)‘𝑦))
9291mpteq2dv 4982 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)))
93 fveq2 6448 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
9492, 93breq12d 4901 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)))
9594rspccva 3510 . . . . . . . . 9 ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
9632, 95sylan 575 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
97 climuni 14700 . . . . . . . 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 579 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = (𝐹𝑦))
9958, 98syl5eqr 2828 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ) = (𝐹𝑦))
10099mpteq2dva 4981 . . . . 5 (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
10154, 100eqtr4d 2817 . . . 4 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )))
102101, 9syl6eqr 2832 . . 3 (𝜑𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < )))
103102fveq2d 6452 . 2 (𝜑 → (∫2𝐹) = (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))))
104 itg2itg1 23951 . . . . . . . 8 (((𝑃𝑚) ∈ dom ∫1 ∧ 0𝑝𝑟 ≤ (𝑃𝑚)) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
10511, 23, 104syl2anc 579 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
106105mpteq2dva 4981 . . . . . 6 (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚))))
107 itg2i1fseq.6 . . . . . 6 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))
108106, 107syl6reqr 2833 . . . . 5 (𝜑𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))))
109108, 50syl6eqr 2832 . . . 4 (𝜑𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
110109rneqd 5600 . . 3 (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
111110supeq1d 8642 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
11253, 103, 1113eqtr4d 2824 1 (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  Vcvv 3398   class class class wbr 4888   ↦ cmpt 4967  dom cdm 5357  ran crn 5358   Fn wfn 6132  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924   ∘𝑟 cofr 7175  supcsup 8636  ℝcr 10273  0cc0 10274  1c1 10275   + caddc 10277  +∞cpnf 10410  ℝ*cxr 10412   < clt 10413   ≤ cle 10414  ℕcn 11379  [,)cico 12494   ⇝ cli 14632  MblFncmbf 23829  ∫1citg1 23830  ∫2citg2 23831  0𝑝c0p 23884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cc 9594  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352  ax-addf 10353 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-disj 4857  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-ofr 7177  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-omul 7850  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-fi 8607  df-sup 8638  df-inf 8639  df-oi 8706  df-card 9100  df-acn 9103  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-z 11734  df-uz 11998  df-q 12101  df-rp 12143  df-xneg 12262  df-xadd 12263  df-xmul 12264  df-ioo 12496  df-ioc 12497  df-ico 12498  df-icc 12499  df-fz 12649  df-fzo 12790  df-fl 12917  df-seq 13125  df-exp 13184  df-hash 13442  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-clim 14636  df-rlim 14637  df-sum 14834  df-rest 16480  df-topgen 16501  df-psmet 20145  df-xmet 20146  df-met 20147  df-bl 20148  df-mopn 20149  df-top 21117  df-topon 21134  df-bases 21169  df-cmp 21610  df-ovol 23679  df-vol 23680  df-mbf 23834  df-itg1 23835  df-itg2 23836  df-0p 23885 This theorem is referenced by:  itg2i1fseq2  23971
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