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Theorem itg2mono 25788
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.3 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
2 rge0ssre 13496 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
3 fss 6752 . . . . . . . . . . . 12 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
41, 2, 3sylancl 586 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
54ffvelcdmda 7104 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
65an32s 652 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
76fmpttd 7135 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
87frnd 6744 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
9 1nn 12277 . . . . . . . . . 10 1 ∈ ℕ
10 eqid 2737 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
1110, 6dmmptd 6713 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
129, 11eleqtrrid 2848 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
1312ne0d 4342 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
14 dm0rn0 5935 . . . . . . . . 9 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
1514necon3bii 2993 . . . . . . . 8 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
1613, 15sylib 218 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
17 itg2mono.5 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
187ffnd 6737 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
19 breq1 5146 . . . . . . . . . . . 12 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
2019ralrn 7108 . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
2322fveq1d 6908 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
24 fvex 6919 . . . . . . . . . . . . . 14 ((𝐹𝑚)‘𝑥) ∈ V
2523, 10, 24fvmpt 7016 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
2625breq1d 5153 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
2726ralbiia 3091 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
2823breq1d 5153 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
2928cbvralvw 3237 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
3027, 29bitr4i 278 . . . . . . . . . 10 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
3121, 30bitrdi 287 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
3231rexbidv 3179 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
3317, 32mpbird 257 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
348, 16, 33suprcld 12231 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
3534rexrd 11311 . . . . 5 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
36 0red 11264 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
37 fveq2 6906 . . . . . . . . . . 11 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
3837feq1d 6720 . . . . . . . . . 10 (𝑛 = 1 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
391ralrimiva 3146 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
409a1i 11 . . . . . . . . . 10 (𝜑 → 1 ∈ ℕ)
4138, 39, 40rspcdva 3623 . . . . . . . . 9 (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
4241ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
43 elrege0 13494 . . . . . . . 8 (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
4442, 43sylib 218 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
4544simpld 494 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
4644simprd 495 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
4737fveq1d 6908 . . . . . . . . . 10 (𝑛 = 1 → ((𝐹𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
48 fvex 6919 . . . . . . . . . 10 ((𝐹‘1)‘𝑥) ∈ V
4947, 10, 48fvmpt 7016 . . . . . . . . 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 7100 . . . . . . . . 9 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
5218, 9, 51sylancl 586 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
5350, 52eqeltrrid 2846 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
548, 16, 33, 53suprubd 12230 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
5536, 45, 34, 46, 54letrd 11418 . . . . 5 ((𝜑𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
56 elxrge0 13497 . . . . 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 583 . . . 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 7134 . . 3 (𝜑𝐺:ℝ⟶(0[,]+∞))
60 itg2cl 25767 . . 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 13476 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
64 fss 6752 . . . . . . . 8 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
651, 63, 64sylancl 586 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
66 itg2cl 25767 . . . . . . 7 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
6765, 66syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
6867fmpttd 7135 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
6968frnd 6744 . . . 4 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
70 supxrcl 13357 . . . 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 2845 . 2 (𝜑𝑆 ∈ ℝ*)
73 itg2mono.2 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
7473adantlr 715 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
751adantlr 715 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
76 itg2mono.4 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
7776adantlr 715 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
7817adantlr 715 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
79 simprll 779 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
80 simprlr 780 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → 𝑓r𝐺)
81 simprr 773 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → ¬ (∫1𝑓) ≤ 𝑆)
8258, 74, 75, 77, 78, 62, 79, 80, 81itg2monolem3 25787 . . . . . . 7 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓r𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → (∫1𝑓) ≤ 𝑆)
8382expr 456 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓r𝐺)) → (¬ (∫1𝑓) ≤ 𝑆 → (∫1𝑓) ≤ 𝑆))
8483pm2.18d 127 . . . . 5 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓r𝐺)) → (∫1𝑓) ≤ 𝑆)
8584expr 456 . . . 4 ((𝜑𝑓 ∈ dom ∫1) → (𝑓r𝐺 → (∫1𝑓) ≤ 𝑆))
8685ralrimiva 3146 . . 3 (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓r𝐺 → (∫1𝑓) ≤ 𝑆))
87 itg2leub 25769 . . . 4 ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓r𝐺 → (∫1𝑓) ≤ 𝑆)))
8859, 72, 87syl2anc 584 . . 3 (𝜑 → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓r𝐺 → (∫1𝑓) ≤ 𝑆)))
8986, 88mpbird 257 . 2 (𝜑 → (∫2𝐺) ≤ 𝑆)
9022feq1d 6720 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑚):ℝ⟶(0[,)+∞)))
9190cbvralvw 3237 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9239, 91sylib 218 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9392r19.21bi 3251 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,)+∞))
94 fss 6752 . . . . . . . 8 (((𝐹𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑚):ℝ⟶(0[,]+∞))
9593, 63, 94sylancl 586 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,]+∞))
9659adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
978, 16, 333jca 1129 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦))
9897adantlr 715 . . . . . . . . . . . 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 727 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
10018adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
101 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
102 fnfvelrn 7100 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
103100, 101, 102syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
10499, 103eqeltrrd 2842 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
105 suprub 12229 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
107 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
108 ltso 11341 . . . . . . . . . . . . 13 < Or ℝ
109108supex 9503 . . . . . . . . . . . 12 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V
11058fvmpt2 7027 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
111107, 109, 110sylancl 586 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
112106, 111breqtrrd 5171 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
113112ralrimiva 3146 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
114 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑧))
115 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
116114, 115breq12d 5156 . . . . . . . . . 10 (𝑥 = 𝑧 → (((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
117116cbvralvw 3237 . . . . . . . . 9 (∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
118113, 117sylib 218 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
11993ffnd 6737 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) Fn ℝ)
12034, 58fmptd 7134 . . . . . . . . . . 11 (𝜑𝐺:ℝ⟶ℝ)
121120ffnd 6737 . . . . . . . . . 10 (𝜑𝐺 Fn ℝ)
122121adantr 480 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
123 reex 11246 . . . . . . . . . 10 ℝ ∈ V
124123a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
125 inidm 4227 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
126 eqidd 2738 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑚)‘𝑧) = ((𝐹𝑚)‘𝑧))
127 eqidd 2738 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
128119, 122, 124, 124, 125, 126, 127ofrfval 7707 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐹𝑚) ∘r𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
129118, 128mpbird 257 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) ∘r𝐺)
130 itg2le 25774 . . . . . . 7 (((𝐹𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹𝑚) ∘r𝐺) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
13195, 96, 129, 130syl3anc 1373 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
132131ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
13368ffnd 6737 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ)
134 breq1 5146 . . . . . . . 8 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) → (𝑧 ≤ (∫2𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
135134ralrn 7108 . . . . . . 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 6911 . . . . . . . . 9 (𝑛 = 𝑚 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑚)))
138 eqid 2737 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
139 fvex 6919 . . . . . . . . 9 (∫2‘(𝐹𝑚)) ∈ V
140137, 138, 139fvmpt 7016 . . . . . . . 8 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) = (∫2‘(𝐹𝑚)))
141140breq1d 5153 . . . . . . 7 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
142141ralbiia 3091 . . . . . 6 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
143136, 142bitrdi 287 . . . . 5 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
144132, 143mpbird 257 . . . 4 (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺))
145 supxrleub 13368 . . . . 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 584 . . . 4 (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺)))
147144, 146mpbird 257 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺))
14862, 147eqbrtrid 5178 . 2 (𝜑𝑆 ≤ (∫2𝐺))
14961, 72, 89, 148xrletrid 13197 1 (𝜑 → (∫2𝐺) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  wss 3951  c0 4333   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  r cofr 7696  supcsup 9480  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cn 12266  [,)cico 13389  [,]cicc 13390  MblFncmbf 25649  1citg1 25650  2citg2 25651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655  df-itg2 25656
This theorem is referenced by:  itg2i1fseq  25790  itg2cnlem1  25796
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