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Theorem itg2mono 23514
 Description: The Monotone Convergence Theorem for nonnegative functions. If {(𝐹‘𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2‘𝐺) is the limit of the sequence {(∫2‘(𝐹‘𝑛)):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 751 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
4 itg2mono.3 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
54adantlr 751 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
6 itg2mono.4 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
76adantlr 751 . . . . . . . 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 751 . . . . . . . 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 802 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
12 simprlr 803 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓𝑟𝐺)
13 simprr 796 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → ¬ (∫1𝑓) ≤ 𝑆)
141, 3, 5, 7, 9, 10, 11, 12, 13itg2monolem3 23513 . . . . . . 7 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → (∫1𝑓) ≤ 𝑆)
1514expr 643 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓𝑟𝐺)) → (¬ (∫1𝑓) ≤ 𝑆 → (∫1𝑓) ≤ 𝑆))
1615pm2.18d 124 . . . . 5 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓𝑟𝐺)) → (∫1𝑓) ≤ 𝑆)
1716expr 643 . . . 4 ((𝜑𝑓 ∈ dom ∫1) → (𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆))
1817ralrimiva 2965 . . 3 (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆))
19 rge0ssre 12277 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
20 fss 6054 . . . . . . . . . . . . 13 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
214, 19, 20sylancl 694 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
2221ffvelrnda 6357 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
2322an32s 846 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
24 eqid 2621 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
2523, 24fmptd 6383 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
26 frn 6051 . . . . . . . . 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 11028 . . . . . . . . . . 11 1 ∈ ℕ
2924, 23dmmptd 6022 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
3028, 29syl5eleqr 2707 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
31 ne0i 3919 . . . . . . . . . 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 5340 . . . . . . . . . 10 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
3433necon3bii 2845 . . . . . . . . 9 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
3532, 34sylib 208 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
36 ffn 6043 . . . . . . . . . . . . 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 4654 . . . . . . . . . . . . 13 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
3938ralrn 6360 . . . . . . . . . . . 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 6189 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
4241fveq1d 6191 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
43 fvex 6199 . . . . . . . . . . . . . . 15 ((𝐹𝑚)‘𝑥) ∈ V
4442, 24, 43fvmpt 6280 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
4544breq1d 4661 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
4645ralbiia 2978 . . . . . . . . . . . 12 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
4742breq1d 4661 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
4847cbvralv 3169 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
4946, 48bitr4i 267 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
5040, 49syl6bb 276 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
5150rexbidv 3050 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
528, 51mpbird 247 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
53 suprcl 10980 . . . . . . . 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 1325 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
5554rexrd 10086 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
56 0red 10038 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
574ralrimiva 2965 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
58 fveq2 6189 . . . . . . . . . . . . 13 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
5958feq1d 6028 . . . . . . . . . . . 12 (𝑛 = 1 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
6059rspcv 3303 . . . . . . . . . . 11 (1 ∈ ℕ → (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) → (𝐹‘1):ℝ⟶(0[,)+∞)))
6128, 57, 60mpsyl 68 . . . . . . . . . 10 (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
6261ffvelrnda 6357 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
63 elrege0 12275 . . . . . . . . 9 (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
6462, 63sylib 208 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
6564simpld 475 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
6664simprd 479 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
6758fveq1d 6191 . . . . . . . . . . 11 (𝑛 = 1 → ((𝐹𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
68 fvex 6199 . . . . . . . . . . 11 ((𝐹‘1)‘𝑥) ∈ V
6967, 24, 68fvmpt 6280 . . . . . . . . . 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 6354 . . . . . . . . . 10 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
7237, 28, 71sylancl 694 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
7370, 72syl5eqelr 2705 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
74 suprub 10981 . . . . . . . 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 1328 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
7656, 65, 54, 66, 75letrd 10191 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
77 elxrge0 12278 . . . . . 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 698 . . . . 5 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞))
7978, 1fmptd 6383 . . . 4 (𝜑𝐺:ℝ⟶(0[,]+∞))
80 icossicc 12257 . . . . . . . . . 10 (0[,)+∞) ⊆ (0[,]+∞)
81 fss 6054 . . . . . . . . . 10 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
824, 80, 81sylancl 694 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
83 itg2cl 23493 . . . . . . . . 9 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
8482, 83syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
85 eqid 2621 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
8684, 85fmptd 6383 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
87 frn 6051 . . . . . . 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 12142 . . . . . 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 2704 . . . 4 (𝜑𝑆 ∈ ℝ*)
92 itg2leub 23495 . . . 4 ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆)))
9379, 91, 92syl2anc 693 . . 3 (𝜑 → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆)))
9418, 93mpbird 247 . 2 (𝜑 → (∫2𝐺) ≤ 𝑆)
9541feq1d 6028 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑚):ℝ⟶(0[,)+∞)))
9695cbvralv 3169 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9757, 96sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9897r19.21bi 2931 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,)+∞))
99 fss 6054 . . . . . . . 8 (((𝐹𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑚):ℝ⟶(0[,]+∞))
10098, 80, 99sylancl 694 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,]+∞))
10179adantr 481 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
10227, 35, 523jca 1241 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦))
103102adantlr 751 . . . . . . . . . . . 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 763 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
10537adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
106 simplr 792 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
107 fnfvelrn 6354 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
108105, 106, 107syl2anc 693 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
109104, 108eqeltrrd 2701 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
110 suprub 10981 . . . . . . . . . . . 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 693 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
112 simpr 477 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
113 ltso 10115 . . . . . . . . . . . . 13 < Or ℝ
114113supex 8366 . . . . . . . . . . . 12 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V
1151fvmpt2 6289 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
116112, 114, 115sylancl 694 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
117111, 116breqtrrd 4679 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
118117ralrimiva 2965 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
119 fveq2 6189 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑧))
120 fveq2 6189 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
121119, 120breq12d 4664 . . . . . . . . . 10 (𝑥 = 𝑧 → (((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
122121cbvralv 3169 . . . . . . . . 9 (∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
123118, 122sylib 208 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
124 ffn 6043 . . . . . . . . . 10 ((𝐹𝑚):ℝ⟶(0[,]+∞) → (𝐹𝑚) Fn ℝ)
125100, 124syl 17 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) Fn ℝ)
12654, 1fmptd 6383 . . . . . . . . . . 11 (𝜑𝐺:ℝ⟶ℝ)
127 ffn 6043 . . . . . . . . . . 11 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
128126, 127syl 17 . . . . . . . . . 10 (𝜑𝐺 Fn ℝ)
129128adantr 481 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
130 reex 10024 . . . . . . . . . 10 ℝ ∈ V
131130a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
132 inidm 3820 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
133 eqidd 2622 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑚)‘𝑧) = ((𝐹𝑚)‘𝑧))
134 eqidd 2622 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
135125, 129, 131, 131, 132, 133, 134ofrfval 6902 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐹𝑚) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
136123, 135mpbird 247 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) ∘𝑟𝐺)
137 itg2le 23500 . . . . . . 7 (((𝐹𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹𝑚) ∘𝑟𝐺) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
138100, 101, 136, 137syl3anc 1325 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
139138ralrimiva 2965 . . . . 5 (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
140 ffn 6043 . . . . . . . 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 4654 . . . . . . . 8 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) → (𝑧 ≤ (∫2𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
143142ralrn 6360 . . . . . . 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 6193 . . . . . . . . 9 (𝑛 = 𝑚 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑚)))
146 fvex 6199 . . . . . . . . 9 (∫2‘(𝐹𝑚)) ∈ V
147145, 85, 146fvmpt 6280 . . . . . . . 8 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) = (∫2‘(𝐹𝑚)))
148147breq1d 4661 . . . . . . 7 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
149148ralbiia 2978 . . . . . 6 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
150144, 149syl6bb 276 . . . . 5 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
151139, 150mpbird 247 . . . 4 (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺))
152 itg2cl 23493 . . . . . 6 (𝐺:ℝ⟶(0[,]+∞) → (∫2𝐺) ∈ ℝ*)
15379, 152syl 17 . . . . 5 (𝜑 → (∫2𝐺) ∈ ℝ*)
154 supxrleub 12153 . . . . 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 693 . . . 4 (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺)))
156151, 155mpbird 247 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺))
15710, 156syl5eqbr 4686 . 2 (𝜑𝑆 ≤ (∫2𝐺))
158 xrletri3 11982 . . 3 (((∫2𝐺) ∈ ℝ*𝑆 ∈ ℝ*) → ((∫2𝐺) = 𝑆 ↔ ((∫2𝐺) ≤ 𝑆𝑆 ≤ (∫2𝐺))))
159153, 91, 158syl2anc 693 . 2 (𝜑 → ((∫2𝐺) = 𝑆 ↔ ((∫2𝐺) ≤ 𝑆𝑆 ≤ (∫2𝐺))))
16094, 157, 159mpbir2and 957 1 (𝜑 → (∫2𝐺) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1482   ∈ wcel 1989   ≠ wne 2793  ∀wral 2911  ∃wrex 2912  Vcvv 3198   ⊆ wss 3572  ∅c0 3913   class class class wbr 4651   ↦ cmpt 4727  dom cdm 5112  ran crn 5113   Fn wfn 5881  ⟶wf 5882  ‘cfv 5886  (class class class)co 6647   ∘𝑟 cofr 6893  supcsup 8343  ℝcr 9932  0cc0 9933  1c1 9934   + caddc 9936  +∞cpnf 10068  ℝ*cxr 10070   < clt 10071   ≤ cle 10072  ℕcn 11017  [,)cico 12174  [,]cicc 12175  MblFncmbf 23377  ∫1citg1 23378  ∫2citg2 23379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cc 9254  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011  ax-addf 10012 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-disj 4619  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894  df-ofr 6895  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-omul 7562  df-er 7739  df-map 7856  df-pm 7857  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fi 8314  df-sup 8345  df-inf 8346  df-oi 8412  df-card 8762  df-acn 8765  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-n0 11290  df-z 11375  df-uz 11685  df-q 11786  df-rp 11830  df-xneg 11943  df-xadd 11944  df-xmul 11945  df-ioo 12176  df-ioc 12177  df-ico 12178  df-icc 12179  df-fz 12324  df-fzo 12462  df-fl 12588  df-seq 12797  df-exp 12856  df-hash 13113  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-clim 14213  df-rlim 14214  df-sum 14411  df-rest 16077  df-topgen 16098  df-psmet 19732  df-xmet 19733  df-met 19734  df-bl 19735  df-mopn 19736  df-top 20693  df-topon 20710  df-bases 20744  df-cmp 21184  df-ovol 23227  df-vol 23228  df-mbf 23382  df-itg1 23383  df-itg2 23384 This theorem is referenced by:  itg2i1fseq  23516  itg2cnlem1  23522
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