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Theorem itg2mono 24453
 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.3 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
2 rge0ssre 12888 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
3 fss 6512 . . . . . . . . . . . 12 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
41, 2, 3sylancl 589 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
54ffvelrnda 6842 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
65an32s 651 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
76fmpttd 6870 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
87frnd 6505 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
9 1nn 11685 . . . . . . . . . 10 1 ∈ ℕ
10 eqid 2758 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
1110, 6dmmptd 6476 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
129, 11eleqtrrid 2859 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
1312ne0d 4234 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
14 dm0rn0 5766 . . . . . . . . 9 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
1514necon3bii 3003 . . . . . . . 8 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
1613, 15sylib 221 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
17 itg2mono.5 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
187ffnd 6499 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
19 breq1 5035 . . . . . . . . . . . 12 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
2019ralrn 6845 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
2118, 20syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
22 fveq2 6658 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
2322fveq1d 6660 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
24 fvex 6671 . . . . . . . . . . . . . 14 ((𝐹𝑚)‘𝑥) ∈ V
2523, 10, 24fvmpt 6759 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
2625breq1d 5042 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
2726ralbiia 3096 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
2823breq1d 5042 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
2928cbvralvw 3361 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
3027, 29bitr4i 281 . . . . . . . . . 10 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
3121, 30bitrdi 290 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
3231rexbidv 3221 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
3317, 32mpbird 260 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
348, 16, 33suprcld 11640 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
3534rexrd 10729 . . . . 5 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
36 0red 10682 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
37 fveq2 6658 . . . . . . . . . . 11 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
3837feq1d 6483 . . . . . . . . . 10 (𝑛 = 1 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
391ralrimiva 3113 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
409a1i 11 . . . . . . . . . 10 (𝜑 → 1 ∈ ℕ)
4138, 39, 40rspcdva 3543 . . . . . . . . 9 (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
4241ffvelrnda 6842 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
43 elrege0 12886 . . . . . . . 8 (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
4442, 43sylib 221 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
4544simpld 498 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
4644simprd 499 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
4737fveq1d 6660 . . . . . . . . . 10 (𝑛 = 1 → ((𝐹𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
48 fvex 6671 . . . . . . . . . 10 ((𝐹‘1)‘𝑥) ∈ V
4947, 10, 48fvmpt 6759 . . . . . . . . 9 (1 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥))
509, 49ax-mp 5 . . . . . . . 8 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥)
51 fnfvelrn 6839 . . . . . . . . 9 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
5218, 9, 51sylancl 589 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
5350, 52eqeltrrid 2857 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
548, 16, 33, 53suprubd 11639 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
5536, 45, 34, 46, 54letrd 10835 . . . . 5 ((𝜑𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
56 elxrge0 12889 . . . . 5 (sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞) ↔ (sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ* ∧ 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < )))
5735, 55, 56sylanbrc 586 . . . 4 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞))
58 itg2mono.1 . . . 4 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
5957, 58fmptd 6869 . . 3 (𝜑𝐺:ℝ⟶(0[,]+∞))
60 itg2cl 24432 . . 3 (𝐺:ℝ⟶(0[,]+∞) → (∫2𝐺) ∈ ℝ*)
6159, 60syl 17 . 2 (𝜑 → (∫2𝐺) ∈ ℝ*)
62 itg2mono.6 . . 3 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )
63 icossicc 12868 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
64 fss 6512 . . . . . . . 8 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
651, 63, 64sylancl 589 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
66 itg2cl 24432 . . . . . . 7 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
6765, 66syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
6867fmpttd 6870 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
6968frnd 6505 . . . 4 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
70 supxrcl 12749 . . . 4 (ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ* → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ∈ ℝ*)
7169, 70syl 17 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ∈ ℝ*)
7262, 71eqeltrid 2856 . 2 (𝜑𝑆 ∈ ℝ*)
73 itg2mono.2 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
7473adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
751adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
76 itg2mono.4 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
7776adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
7817adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
79 simprll 778 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
80 simprlr 779 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓r𝐺)
81 simprr 772 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → ¬ (∫1𝑓) ≤ 𝑆)
8258, 74, 75, 77, 78, 62, 79, 80, 81itg2monolem3 24452 . . . . . . 7 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → (∫1𝑓) ≤ 𝑆)
8382expr 460 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓r𝐺)) → (¬ (∫1𝑓) ≤ 𝑆 → (∫1𝑓) ≤ 𝑆))
8483pm2.18d 127 . . . . 5 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓r𝐺)) → (∫1𝑓) ≤ 𝑆)
8584expr 460 . . . 4 ((𝜑𝑓 ∈ dom ∫1) → (𝑓r𝐺 → (∫1𝑓) ≤ 𝑆))
8685ralrimiva 3113 . . 3 (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓r𝐺 → (∫1𝑓) ≤ 𝑆))
87 itg2leub 24434 . . . 4 ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓r𝐺 → (∫1𝑓) ≤ 𝑆)))
8859, 72, 87syl2anc 587 . . 3 (𝜑 → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓r𝐺 → (∫1𝑓) ≤ 𝑆)))
8986, 88mpbird 260 . 2 (𝜑 → (∫2𝐺) ≤ 𝑆)
9022feq1d 6483 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑚):ℝ⟶(0[,)+∞)))
9190cbvralvw 3361 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9239, 91sylib 221 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9392r19.21bi 3137 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,)+∞))
94 fss 6512 . . . . . . . 8 (((𝐹𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑚):ℝ⟶(0[,]+∞))
9593, 63, 94sylancl 589 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,]+∞))
9659adantr 484 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
978, 16, 333jca 1125 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦))
9897adantlr 714 . . . . . . . . . . . 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))
9925ad2antlr 726 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
10018adantlr 714 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
101 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
102 fnfvelrn 6839 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
103100, 101, 102syl2anc 587 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
10499, 103eqeltrrd 2853 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
105 suprub 11638 . . . . . . . . . . . 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, < ))
10698, 104, 105syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
107 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
108 ltso 10759 . . . . . . . . . . . . 13 < Or ℝ
109108supex 8960 . . . . . . . . . . . 12 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V
11058fvmpt2 6770 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
111107, 109, 110sylancl 589 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
112106, 111breqtrrd 5060 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
113112ralrimiva 3113 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
114 fveq2 6658 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑧))
115 fveq2 6658 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
116114, 115breq12d 5045 . . . . . . . . . 10 (𝑥 = 𝑧 → (((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
117116cbvralvw 3361 . . . . . . . . 9 (∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
118113, 117sylib 221 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
11993ffnd 6499 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) Fn ℝ)
12034, 58fmptd 6869 . . . . . . . . . . 11 (𝜑𝐺:ℝ⟶ℝ)
121120ffnd 6499 . . . . . . . . . 10 (𝜑𝐺 Fn ℝ)
122121adantr 484 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
123 reex 10666 . . . . . . . . . 10 ℝ ∈ V
124123a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
125 inidm 4123 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
126 eqidd 2759 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑚)‘𝑧) = ((𝐹𝑚)‘𝑧))
127 eqidd 2759 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
128119, 122, 124, 124, 125, 126, 127ofrfval 7414 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐹𝑚) ∘r𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
129118, 128mpbird 260 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) ∘r𝐺)
130 itg2le 24439 . . . . . . 7 (((𝐹𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹𝑚) ∘r𝐺) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
13195, 96, 129, 130syl3anc 1368 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
132131ralrimiva 3113 . . . . 5 (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
13368ffnd 6499 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ)
134 breq1 5035 . . . . . . . 8 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) → (𝑧 ≤ (∫2𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
135134ralrn 6845 . . . . . . 7 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
136133, 135syl 17 . . . . . 6 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
137 2fveq3 6663 . . . . . . . . 9 (𝑛 = 𝑚 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑚)))
138 eqid 2758 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
139 fvex 6671 . . . . . . . . 9 (∫2‘(𝐹𝑚)) ∈ V
140137, 138, 139fvmpt 6759 . . . . . . . 8 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) = (∫2‘(𝐹𝑚)))
141140breq1d 5042 . . . . . . 7 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
142141ralbiia 3096 . . . . . 6 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
143136, 142bitrdi 290 . . . . 5 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
144132, 143mpbird 260 . . . 4 (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺))
145 supxrleub 12760 . . . . 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)))
14669, 61, 145syl2anc 587 . . . 4 (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺)))
147144, 146mpbird 260 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺))
14862, 147eqbrtrid 5067 . 2 (𝜑𝑆 ≤ (∫2𝐺))
14961, 72, 89, 148xrletrid 12589 1 (𝜑 → (∫2𝐺) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3858  ∅c0 4225   class class class wbr 5032   ↦ cmpt 5112  dom cdm 5524  ran crn 5525   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∘r cofr 7404  supcsup 8937  ℝcr 10574  0cc0 10575  1c1 10576   + caddc 10578  +∞cpnf 10710  ℝ*cxr 10712   < clt 10713   ≤ cle 10714  ℕcn 11674  [,)cico 12781  [,]cicc 12782  MblFncmbf 24314  ∫1citg1 24315  ∫2citg2 24316 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-inf2 9137  ax-cc 9895  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653  ax-addf 10654 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-disj 4998  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7405  df-ofr 7406  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-oadd 8116  df-omul 8117  df-er 8299  df-map 8418  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-fi 8908  df-sup 8939  df-inf 8940  df-oi 9007  df-dju 9363  df-card 9401  df-acn 9404  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-3 11738  df-n0 11935  df-z 12021  df-uz 12283  df-q 12389  df-rp 12431  df-xneg 12548  df-xadd 12549  df-xmul 12550  df-ioo 12783  df-ioc 12784  df-ico 12785  df-icc 12786  df-fz 12940  df-fzo 13083  df-fl 13211  df-seq 13419  df-exp 13480  df-hash 13741  df-cj 14506  df-re 14507  df-im 14508  df-sqrt 14642  df-abs 14643  df-clim 14893  df-rlim 14894  df-sum 15091  df-rest 16754  df-topgen 16775  df-psmet 20158  df-xmet 20159  df-met 20160  df-bl 20161  df-mopn 20162  df-top 21594  df-topon 21611  df-bases 21646  df-cmp 22087  df-ovol 24164  df-vol 24165  df-mbf 24319  df-itg1 24320  df-itg2 24321 This theorem is referenced by:  itg2i1fseq  24455  itg2cnlem1  24461
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