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Theorem itg2i1fseq 23642
Description: Subject to the conditions coming from mbfi1fseq 23608, 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 6304 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃𝑛) = (𝑃𝑚))
21fveq1d 6306 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑚)‘𝑥))
32cbvmptv 4858 . . . . . . 7 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥))
4 fveq2 6304 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑚)‘𝑥) = ((𝑃𝑚)‘𝑦))
54mpteq2dv 4853 . . . . . . 7 (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
63, 5syl5eq 2770 . . . . . 6 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
76rneqd 5460 . . . . 5 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
87supeq1d 8468 . . . 4 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ))
98cbvmptv 4858 . . 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 6474 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ dom ∫1)
12 i1fmbf 23562 . . . 4 ((𝑃𝑚) ∈ dom ∫1 → (𝑃𝑚) ∈ MblFn)
1311, 12syl 17 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ MblFn)
14 i1ff 23563 . . . . 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 4772 . . . . . . . 8 (𝑛 = 𝑚 → (0𝑝𝑟 ≤ (𝑃𝑛) ↔ 0𝑝𝑟 ≤ (𝑃𝑚)))
18 oveq1 6772 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
1918fveq2d 6308 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
201, 19breq12d 4773 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))))
2117, 20anbi12d 749 . . . . . . 7 (𝑛 = 𝑚 → ((0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝𝑟 ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)))))
2221rspccva 3412 . . . . . 6 ((∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) ∧ 𝑚 ∈ ℕ) → (0𝑝𝑟 ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))))
2316, 22sylan 489 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (0𝑝𝑟 ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))))
2423simpld 477 . . . 4 ((𝜑𝑚 ∈ ℕ) → 0𝑝𝑟 ≤ (𝑃𝑚))
25 0plef 23559 . . . 4 ((𝑃𝑚):ℝ⟶(0[,)+∞) ↔ ((𝑃𝑚):ℝ⟶ℝ ∧ 0𝑝𝑟 ≤ (𝑃𝑚)))
2615, 24, 25sylanbrc 701 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚):ℝ⟶(0[,)+∞))
2723simprd 482 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)))
28 rge0ssre 12394 . . . . 5 (0[,)+∞) ⊆ ℝ
29 itg2i1fseq.2 . . . . . 6 (𝜑𝐹:ℝ⟶(0[,)+∞))
3029ffvelrnda 6474 . . . . 5 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
3128, 30sseldi 3707 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
32 itg2i1fseq.1 . . . . . . . . 9 (𝜑𝐹 ∈ MblFn)
33 itg2i1fseq.5 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
3432, 29, 10, 16, 33itg2i1fseqle 23641 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘𝑟𝐹)
35 ffn 6158 . . . . . . . . . 10 ((𝑃𝑚):ℝ⟶ℝ → (𝑃𝑚) Fn ℝ)
3615, 35syl 17 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) Fn ℝ)
37 ffn 6158 . . . . . . . . . . 11 (𝐹:ℝ⟶(0[,)+∞) → 𝐹 Fn ℝ)
3829, 37syl 17 . . . . . . . . . 10 (𝜑𝐹 Fn ℝ)
3938adantr 472 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
40 reex 10140 . . . . . . . . . 10 ℝ ∈ V
4140a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
42 inidm 3930 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
43 eqidd 2725 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) = ((𝑃𝑚)‘𝑦))
44 eqidd 2725 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
4536, 39, 41, 41, 42, 43, 44ofrfval 7022 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘𝑟𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)))
4634, 45mpbid 222 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4746r19.21bi 3034 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4847an32s 881 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4948ralrimiva 3068 . . . 4 ((𝜑𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
50 breq2 4764 . . . . . 6 (𝑧 = (𝐹𝑦) → (((𝑃𝑚)‘𝑦) ≤ 𝑧 ↔ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)))
5150ralbidv 3088 . . . . 5 (𝑧 = (𝐹𝑦) → (∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)))
5251rspcev 3413 . . . 4 (((𝐹𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
5331, 49, 52syl2anc 696 . . 3 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
541fveq2d 6308 . . . . . 6 (𝑛 = 𝑚 → (∫2‘(𝑃𝑛)) = (∫2‘(𝑃𝑚)))
5554cbvmptv 4858 . . . . 5 (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5655rneqi 5459 . . . 4 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5756supeq1i 8469 . . 3 sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))), ℝ*, < )
589, 13, 26, 27, 53, 57itg2mono 23640 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
5929feqmptd 6363 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
601fveq1d 6306 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑚)‘𝑦))
6160cbvmptv 4858 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
6261rneqi 5459 . . . . . . . 8 ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
6362supeq1i 8469 . . . . . . 7 sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )
64 nnuz 11837 . . . . . . . . 9 ℕ = (ℤ‘1)
65 1zzd 11521 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 1 ∈ ℤ)
6615ffvelrnda 6474 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6766an32s 881 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6867, 61fmptd 6500 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)):ℕ⟶ℝ)
69 peano2nn 11145 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
70 ffvelrn 6472 . . . . . . . . . . . . . . . . 17 ((𝑃:ℕ⟶dom ∫1 ∧ (𝑚 + 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
7110, 69, 70syl2an 495 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
72 i1ff 23563 . . . . . . . . . . . . . . . 16 ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
7371, 72syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
74 ffn 6158 . . . . . . . . . . . . . . 15 ((𝑃‘(𝑚 + 1)):ℝ⟶ℝ → (𝑃‘(𝑚 + 1)) Fn ℝ)
7573, 74syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
76 eqidd 2725 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
7736, 75, 41, 41, 42, 43, 76ofrfval 7022 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)))
7827, 77mpbid 222 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7978r19.21bi 3034 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
8079an32s 881 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
81 eqid 2724 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))
82 fvex 6314 . . . . . . . . . . . 12 ((𝑃𝑚)‘𝑦) ∈ V
8360, 81, 82fvmpt 6396 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
8483adantl 473 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
85 fveq2 6304 . . . . . . . . . . . . . 14 (𝑛 = (𝑚 + 1) → (𝑃𝑛) = (𝑃‘(𝑚 + 1)))
8685fveq1d 6306 . . . . . . . . . . . . 13 (𝑛 = (𝑚 + 1) → ((𝑃𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
87 fvex 6314 . . . . . . . . . . . . 13 ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
8886, 81, 87fvmpt 6396 . . . . . . . . . . . 12 ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8969, 88syl 17 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
9089adantl 473 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
9180, 84, 903brtr4d 4792 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)))
9283breq1d 4770 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃𝑚)‘𝑦) ≤ 𝑧))
9392ralbiia 3081 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
9493rexbii 3143 . . . . . . . . . 10 (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
9553, 94sylibr 224 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧)
9664, 65, 68, 91, 95climsup 14520 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ))
97 fveq2 6304 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑛)‘𝑦))
9897mpteq2dv 4853 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)))
99 fveq2 6304 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
10098, 99breq12d 4773 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)))
101100rspccva 3412 . . . . . . . . 9 ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
10233, 101sylan 489 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
103 climuni 14403 . . . . . . . 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))
10496, 102, 103syl2anc 696 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = (𝐹𝑦))
10563, 104syl5eqr 2772 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ) = (𝐹𝑦))
106105mpteq2dva 4852 . . . . 5 (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
10759, 106eqtr4d 2761 . . . 4 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )))
108107, 9syl6eqr 2776 . . 3 (𝜑𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < )))
109108fveq2d 6308 . 2 (𝜑 → (∫2𝐹) = (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))))
110 itg2itg1 23623 . . . . . . . 8 (((𝑃𝑚) ∈ dom ∫1 ∧ 0𝑝𝑟 ≤ (𝑃𝑚)) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
11111, 24, 110syl2anc 696 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
112111mpteq2dva 4852 . . . . . 6 (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚))))
113 itg2i1fseq.6 . . . . . 6 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))
114112, 113syl6reqr 2777 . . . . 5 (𝜑𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))))
115114, 55syl6eqr 2776 . . . 4 (𝜑𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
116115rneqd 5460 . . 3 (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
117116supeq1d 8468 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
11858, 109, 1173eqtr4d 2768 1 (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wral 3014  wrex 3015  Vcvv 3304   class class class wbr 4760  cmpt 4837  dom cdm 5218  ran crn 5219   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  𝑟 cofr 7013  supcsup 8462  cr 10048  0cc0 10049  1c1 10050   + caddc 10052  +∞cpnf 10184  *cxr 10186   < clt 10187  cle 10188  cn 11133  [,)cico 12291  cli 14335  MblFncmbf 23503  1citg1 23504  2citg2 23505  0𝑝c0p 23556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cc 9370  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127  ax-addf 10128
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-disj 4729  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-of 7014  df-ofr 7015  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-omul 7685  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fi 8433  df-sup 8464  df-inf 8465  df-oi 8531  df-card 8878  df-acn 8881  df-cda 9103  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-q 11903  df-rp 11947  df-xneg 12060  df-xadd 12061  df-xmul 12062  df-ioo 12293  df-ioc 12294  df-ico 12295  df-icc 12296  df-fz 12441  df-fzo 12581  df-fl 12708  df-seq 12917  df-exp 12976  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339  df-rlim 14340  df-sum 14537  df-rest 16206  df-topgen 16227  df-psmet 19861  df-xmet 19862  df-met 19863  df-bl 19864  df-mopn 19865  df-top 20822  df-topon 20839  df-bases 20873  df-cmp 21313  df-ovol 23354  df-vol 23355  df-mbf 23508  df-itg1 23509  df-itg2 23510  df-0p 23557
This theorem is referenced by:  itg2i1fseq2  23643
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