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Theorem itg2mono 23243
Description: The Monotone Convergence Theorem for nonnegative functions. If {(𝐹𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2𝐺) is the limit of the sequence {(∫2‘(𝐹𝑛)):𝑛 ∈ ℕ}. (Contributed by Mario Carneiro, 16-Aug-2014.)
Hypotheses
Ref Expression
itg2mono.1 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
itg2mono.2 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
itg2mono.3 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
itg2mono.4 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
itg2mono.5 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
itg2mono.6 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )
Assertion
Ref Expression
itg2mono (𝜑 → (∫2𝐺) = 𝑆)
Distinct variable groups:   𝑥,𝑛,𝑦,𝐺   𝑛,𝐹,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦   𝑆,𝑛,𝑥,𝑦

Proof of Theorem itg2mono
Dummy variables 𝑓 𝑚 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itg2mono.1 . . . . . . . 8 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
2 itg2mono.2 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
32adantlr 746 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
4 itg2mono.3 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
54adantlr 746 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
6 itg2mono.4 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
76adantlr 746 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
8 itg2mono.5 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
98adantlr 746 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
10 itg2mono.6 . . . . . . . 8 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )
11 simprll 797 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
12 simprlr 798 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓𝑟𝐺)
13 simprr 791 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → ¬ (∫1𝑓) ≤ 𝑆)
141, 3, 5, 7, 9, 10, 11, 12, 13itg2monolem3 23242 . . . . . . 7 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → (∫1𝑓) ≤ 𝑆)
1514expr 640 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓𝑟𝐺)) → (¬ (∫1𝑓) ≤ 𝑆 → (∫1𝑓) ≤ 𝑆))
1615pm2.18d 122 . . . . 5 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓𝑟𝐺)) → (∫1𝑓) ≤ 𝑆)
1716expr 640 . . . 4 ((𝜑𝑓 ∈ dom ∫1) → (𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆))
1817ralrimiva 2948 . . 3 (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆))
19 rge0ssre 12107 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
20 fss 5955 . . . . . . . . . . . . 13 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
214, 19, 20sylancl 692 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
2221ffvelrnda 6252 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
2322an32s 841 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
24 eqid 2609 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
2523, 24fmptd 6277 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
26 frn 5952 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
2725, 26syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
28 1nn 10878 . . . . . . . . . . 11 1 ∈ ℕ
2924, 23dmmptd 5923 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
3028, 29syl5eleqr 2694 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
31 ne0i 3879 . . . . . . . . . 10 (1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
3230, 31syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
33 dm0rn0 5250 . . . . . . . . . 10 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
3433necon3bii 2833 . . . . . . . . 9 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
3532, 34sylib 206 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
36 ffn 5944 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
3725, 36syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
38 breq1 4580 . . . . . . . . . . . . 13 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
3938ralrn 6255 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
4037, 39syl 17 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
41 fveq2 6088 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
4241fveq1d 6090 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
43 fvex 6098 . . . . . . . . . . . . . . 15 ((𝐹𝑚)‘𝑥) ∈ V
4442, 24, 43fvmpt 6176 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
4544breq1d 4587 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
4645ralbiia 2961 . . . . . . . . . . . 12 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
4742breq1d 4587 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
4847cbvralv 3146 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
4946, 48bitr4i 265 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
5040, 49syl6bb 274 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
5150rexbidv 3033 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
528, 51mpbird 245 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
53 suprcl 10832 . . . . . . . 8 ((ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
5427, 35, 52, 53syl3anc 1317 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
5554rexrd 9945 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
56 0red 9897 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
574ralrimiva 2948 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
58 fveq2 6088 . . . . . . . . . . . . 13 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
5958feq1d 5929 . . . . . . . . . . . 12 (𝑛 = 1 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
6059rspcv 3277 . . . . . . . . . . 11 (1 ∈ ℕ → (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) → (𝐹‘1):ℝ⟶(0[,)+∞)))
6128, 57, 60mpsyl 65 . . . . . . . . . 10 (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
6261ffvelrnda 6252 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
63 elrege0 12105 . . . . . . . . 9 (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
6462, 63sylib 206 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
6564simpld 473 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
6664simprd 477 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
6758fveq1d 6090 . . . . . . . . . . 11 (𝑛 = 1 → ((𝐹𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
68 fvex 6098 . . . . . . . . . . 11 ((𝐹‘1)‘𝑥) ∈ V
6967, 24, 68fvmpt 6176 . . . . . . . . . 10 (1 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥))
7028, 69ax-mp 5 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥)
71 fnfvelrn 6249 . . . . . . . . . 10 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
7237, 28, 71sylancl 692 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
7370, 72syl5eqelr 2692 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
74 suprub 10833 . . . . . . . 8 (((ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦) ∧ ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
7527, 35, 52, 73, 74syl31anc 1320 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
7656, 65, 54, 66, 75letrd 10045 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
77 elxrge0 12108 . . . . . 6 (sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞) ↔ (sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ* ∧ 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < )))
7855, 76, 77sylanbrc 694 . . . . 5 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞))
7978, 1fmptd 6277 . . . 4 (𝜑𝐺:ℝ⟶(0[,]+∞))
80 icossicc 12087 . . . . . . . . . 10 (0[,)+∞) ⊆ (0[,]+∞)
81 fss 5955 . . . . . . . . . 10 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
824, 80, 81sylancl 692 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
83 itg2cl 23222 . . . . . . . . 9 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
8482, 83syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
85 eqid 2609 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
8684, 85fmptd 6277 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
87 frn 5952 . . . . . . 7 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ* → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
8886, 87syl 17 . . . . . 6 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
89 supxrcl 11973 . . . . . 6 (ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ* → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ∈ ℝ*)
9088, 89syl 17 . . . . 5 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ∈ ℝ*)
9110, 90syl5eqel 2691 . . . 4 (𝜑𝑆 ∈ ℝ*)
92 itg2leub 23224 . . . 4 ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆)))
9379, 91, 92syl2anc 690 . . 3 (𝜑 → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆)))
9418, 93mpbird 245 . 2 (𝜑 → (∫2𝐺) ≤ 𝑆)
9541feq1d 5929 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑚):ℝ⟶(0[,)+∞)))
9695cbvralv 3146 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9757, 96sylib 206 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9897r19.21bi 2915 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,)+∞))
99 fss 5955 . . . . . . . 8 (((𝐹𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑚):ℝ⟶(0[,]+∞))
10098, 80, 99sylancl 692 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,]+∞))
10179adantr 479 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
10227, 35, 523jca 1234 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦))
103102adantlr 746 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦))
10444ad2antlr 758 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
10537adantlr 746 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
106 simplr 787 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
107 fnfvelrn 6249 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
108105, 106, 107syl2anc 690 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
109104, 108eqeltrrd 2688 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
110 suprub 10833 . . . . . . . . . . . 12 (((ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦) ∧ ((𝐹𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))) → ((𝐹𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
111103, 109, 110syl2anc 690 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
112 simpr 475 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
113 ltso 9969 . . . . . . . . . . . . 13 < Or ℝ
114113supex 8229 . . . . . . . . . . . 12 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V
1151fvmpt2 6185 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
116112, 114, 115sylancl 692 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
117111, 116breqtrrd 4605 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
118117ralrimiva 2948 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
119 fveq2 6088 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑧))
120 fveq2 6088 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
121119, 120breq12d 4590 . . . . . . . . . 10 (𝑥 = 𝑧 → (((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
122121cbvralv 3146 . . . . . . . . 9 (∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
123118, 122sylib 206 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
124 ffn 5944 . . . . . . . . . 10 ((𝐹𝑚):ℝ⟶(0[,]+∞) → (𝐹𝑚) Fn ℝ)
125100, 124syl 17 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) Fn ℝ)
12654, 1fmptd 6277 . . . . . . . . . . 11 (𝜑𝐺:ℝ⟶ℝ)
127 ffn 5944 . . . . . . . . . . 11 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
128126, 127syl 17 . . . . . . . . . 10 (𝜑𝐺 Fn ℝ)
129128adantr 479 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
130 reex 9883 . . . . . . . . . 10 ℝ ∈ V
131130a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
132 inidm 3783 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
133 eqidd 2610 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑚)‘𝑧) = ((𝐹𝑚)‘𝑧))
134 eqidd 2610 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
135125, 129, 131, 131, 132, 133, 134ofrfval 6780 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐹𝑚) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
136123, 135mpbird 245 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) ∘𝑟𝐺)
137 itg2le 23229 . . . . . . 7 (((𝐹𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹𝑚) ∘𝑟𝐺) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
138100, 101, 136, 137syl3anc 1317 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
139138ralrimiva 2948 . . . . 5 (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
140 ffn 5944 . . . . . . . 8 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ* → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ)
14186, 140syl 17 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ)
142 breq1 4580 . . . . . . . 8 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) → (𝑧 ≤ (∫2𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
143142ralrn 6255 . . . . . . 7 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
144141, 143syl 17 . . . . . 6 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
14541fveq2d 6092 . . . . . . . . 9 (𝑛 = 𝑚 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑚)))
146 fvex 6098 . . . . . . . . 9 (∫2‘(𝐹𝑚)) ∈ V
147145, 85, 146fvmpt 6176 . . . . . . . 8 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) = (∫2‘(𝐹𝑚)))
148147breq1d 4587 . . . . . . 7 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
149148ralbiia 2961 . . . . . 6 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
150144, 149syl6bb 274 . . . . 5 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
151139, 150mpbird 245 . . . 4 (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺))
152 itg2cl 23222 . . . . . 6 (𝐺:ℝ⟶(0[,]+∞) → (∫2𝐺) ∈ ℝ*)
15379, 152syl 17 . . . . 5 (𝜑 → (∫2𝐺) ∈ ℝ*)
154 supxrleub 11984 . . . . 5 ((ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ* ∧ (∫2𝐺) ∈ ℝ*) → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺)))
15588, 153, 154syl2anc 690 . . . 4 (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺)))
156151, 155mpbird 245 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺))
15710, 156syl5eqbr 4612 . 2 (𝜑𝑆 ≤ (∫2𝐺))
158 xrletri3 11820 . . 3 (((∫2𝐺) ∈ ℝ*𝑆 ∈ ℝ*) → ((∫2𝐺) = 𝑆 ↔ ((∫2𝐺) ≤ 𝑆𝑆 ≤ (∫2𝐺))))
159153, 91, 158syl2anc 690 . 2 (𝜑 → ((∫2𝐺) = 𝑆 ↔ ((∫2𝐺) ≤ 𝑆𝑆 ≤ (∫2𝐺))))
16094, 157, 159mpbir2and 958 1 (𝜑 → (∫2𝐺) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  wss 3539  c0 3873   class class class wbr 4577  cmpt 4637  dom cdm 5028  ran crn 5029   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  𝑟 cofr 6771  supcsup 8206  cr 9791  0cc0 9792  1c1 9793   + caddc 9795  +∞cpnf 9927  *cxr 9929   < clt 9930  cle 9931  cn 10867  [,)cico 12004  [,]cicc 12005  MblFncmbf 23106  1citg1 23107  2citg2 23108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cc 9117  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-disj 4548  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-omul 7429  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-acn 8628  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ioo 12006  df-ioc 12007  df-ico 12008  df-icc 12009  df-fz 12153  df-fzo 12290  df-fl 12410  df-seq 12619  df-exp 12678  df-hash 12935  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013  df-rlim 14014  df-sum 14211  df-rest 15852  df-topgen 15873  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-top 20463  df-bases 20464  df-topon 20465  df-cmp 20942  df-ovol 22957  df-vol 22958  df-mbf 23111  df-itg1 23112  df-itg2 23113
This theorem is referenced by:  itg2i1fseq  23245  itg2cnlem1  23251
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