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Theorem itg2mono 23827
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 706 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
4 itg2mono.3 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
54adantlr 706 . . . . . . . 8 (((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
6 itg2mono.4 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
76adantlr 706 . . . . . . . 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 706 . . . . . . . 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 789 . . . . . . . 8 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → ¬ (∫1𝑓) ≤ 𝑆)
141, 3, 5, 7, 9, 10, 11, 12, 13itg2monolem3 23826 . . . . . . 7 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑓𝑟𝐺) ∧ ¬ (∫1𝑓) ≤ 𝑆)) → (∫1𝑓) ≤ 𝑆)
1514expr 448 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓𝑟𝐺)) → (¬ (∫1𝑓) ≤ 𝑆 → (∫1𝑓) ≤ 𝑆))
1615pm2.18d 127 . . . . 5 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑓𝑟𝐺)) → (∫1𝑓) ≤ 𝑆)
1716expr 448 . . . 4 ((𝜑𝑓 ∈ dom ∫1) → (𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆))
1817ralrimiva 3113 . . 3 (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆))
19 rge0ssre 12491 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
20 fss 6238 . . . . . . . . . . . . 13 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
214, 19, 20sylancl 580 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
2221ffvelrnda 6553 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
2322an32s 642 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
2423fmpttd 6579 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
2524frnd 6232 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
26 1nn 11292 . . . . . . . . . . 11 1 ∈ ℕ
27 eqid 2765 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
2827, 23dmmptd 6204 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
2926, 28syl5eleqr 2851 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
3029ne0d 4088 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
31 dm0rn0 5512 . . . . . . . . . 10 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
3231necon3bii 2989 . . . . . . . . 9 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
3330, 32sylib 209 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
3424ffnd 6226 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
35 breq1 4814 . . . . . . . . . . . . 13 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
3635ralrn 6556 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
3734, 36syl 17 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
38 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
3938fveq1d 6381 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
40 fvex 6392 . . . . . . . . . . . . . . 15 ((𝐹𝑚)‘𝑥) ∈ V
4139, 27, 40fvmpt 6475 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
4241breq1d 4821 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
4342ralbiia 3126 . . . . . . . . . . . 12 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
4439breq1d 4821 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
4544cbvralv 3319 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
4643, 45bitr4i 269 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
4737, 46syl6bb 278 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
4847rexbidv 3199 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
498, 48mpbird 248 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
50 suprcl 11242 . . . . . . . 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)
5125, 33, 49, 50syl3anc 1490 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
5251rexrd 10347 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
53 0red 10301 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
54 fveq2 6379 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
5554feq1d 6210 . . . . . . . . . . 11 (𝑛 = 1 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
564ralrimiva 3113 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
5726a1i 11 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℕ)
5855, 56, 57rspcdva 3468 . . . . . . . . . 10 (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
5958ffvelrnda 6553 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
60 elrege0 12489 . . . . . . . . 9 (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
6159, 60sylib 209 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
6261simpld 488 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
6361simprd 489 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
6454fveq1d 6381 . . . . . . . . . . 11 (𝑛 = 1 → ((𝐹𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
65 fvex 6392 . . . . . . . . . . 11 ((𝐹‘1)‘𝑥) ∈ V
6664, 27, 65fvmpt 6475 . . . . . . . . . 10 (1 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥))
6726, 66ax-mp 5 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥)
68 fnfvelrn 6550 . . . . . . . . . 10 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
6934, 26, 68sylancl 580 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
7067, 69syl5eqelr 2849 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
71 suprub 11243 . . . . . . . 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, < ))
7225, 33, 49, 70, 71syl31anc 1492 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
7353, 62, 51, 63, 72letrd 10453 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
74 elxrge0 12492 . . . . . 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, < )))
7552, 73, 74sylanbrc 578 . . . . 5 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞))
7675, 1fmptd 6578 . . . 4 (𝜑𝐺:ℝ⟶(0[,]+∞))
77 icossicc 12470 . . . . . . . . . 10 (0[,)+∞) ⊆ (0[,]+∞)
78 fss 6238 . . . . . . . . . 10 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
794, 77, 78sylancl 580 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
80 itg2cl 23806 . . . . . . . . 9 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
8179, 80syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
8281fmpttd 6579 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
8382frnd 6232 . . . . . 6 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
84 supxrcl 12354 . . . . . 6 (ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ* → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ∈ ℝ*)
8583, 84syl 17 . . . . 5 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ∈ ℝ*)
8610, 85syl5eqel 2848 . . . 4 (𝜑𝑆 ∈ ℝ*)
87 itg2leub 23808 . . . 4 ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆)))
8876, 86, 87syl2anc 579 . . 3 (𝜑 → ((∫2𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓𝑟𝐺 → (∫1𝑓) ≤ 𝑆)))
8918, 88mpbird 248 . 2 (𝜑 → (∫2𝐺) ≤ 𝑆)
9038feq1d 6210 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑚):ℝ⟶(0[,)+∞)))
9190cbvralv 3319 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9256, 91sylib 209 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝐹𝑚):ℝ⟶(0[,)+∞))
9392r19.21bi 3079 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,)+∞))
94 fss 6238 . . . . . . . 8 (((𝐹𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑚):ℝ⟶(0[,]+∞))
9593, 77, 94sylancl 580 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚):ℝ⟶(0[,]+∞))
9676adantr 472 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
9725, 33, 493jca 1158 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦))
9897adantlr 706 . . . . . . . . . . . 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))
9941ad2antlr 718 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
10034adantlr 706 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
101 simplr 785 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
102 fnfvelrn 6550 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
103100, 101, 102syl2anc 579 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
10499, 103eqeltrrd 2845 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
105 suprub 11243 . . . . . . . . . . . 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 579 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
107 simpr 477 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
108 ltso 10377 . . . . . . . . . . . . 13 < Or ℝ
109108supex 8580 . . . . . . . . . . . 12 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V
1101fvmpt2 6484 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
111107, 109, 110sylancl 580 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
112106, 111breqtrrd 4839 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
113112ralrimiva 3113 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥))
114 fveq2 6379 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑧))
115 fveq2 6379 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
116114, 115breq12d 4824 . . . . . . . . . 10 (𝑥 = 𝑧 → (((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
117116cbvralv 3319 . . . . . . . . 9 (∀𝑥 ∈ ℝ ((𝐹𝑚)‘𝑥) ≤ (𝐺𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
118113, 117sylib 209 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧))
11993ffnd 6226 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) Fn ℝ)
12051, 1fmptd 6578 . . . . . . . . . . 11 (𝜑𝐺:ℝ⟶ℝ)
121120ffnd 6226 . . . . . . . . . 10 (𝜑𝐺 Fn ℝ)
122121adantr 472 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
123 reex 10284 . . . . . . . . . 10 ℝ ∈ V
124123a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
125 inidm 3984 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
126 eqidd 2766 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑚)‘𝑧) = ((𝐹𝑚)‘𝑧))
127 eqidd 2766 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
128119, 122, 124, 124, 125, 126, 127ofrfval 7107 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐹𝑚) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹𝑚)‘𝑧) ≤ (𝐺𝑧)))
129118, 128mpbird 248 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝐹𝑚) ∘𝑟𝐺)
130 itg2le 23813 . . . . . . 7 (((𝐹𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹𝑚) ∘𝑟𝐺) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
13195, 96, 129, 130syl3anc 1490 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
132131ralrimiva 3113 . . . . 5 (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
13382ffnd 6226 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ)
134 breq1 4814 . . . . . . . 8 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) → (𝑧 ≤ (∫2𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺)))
135134ralrn 6556 . . . . . . 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 6384 . . . . . . . . 9 (𝑛 = 𝑚 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑚)))
138 eqid 2765 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
139 fvex 6392 . . . . . . . . 9 (∫2‘(𝐹𝑚)) ∈ V
140137, 138, 139fvmpt 6475 . . . . . . . 8 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) = (∫2‘(𝐹𝑚)))
141140breq1d 4821 . . . . . . 7 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
142141ralbiia 3126 . . . . . 6 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑚) ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺))
143136, 142syl6bb 278 . . . . 5 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹𝑚)) ≤ (∫2𝐺)))
144132, 143mpbird 248 . . . 4 (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺))
145 itg2cl 23806 . . . . . 6 (𝐺:ℝ⟶(0[,]+∞) → (∫2𝐺) ∈ ℝ*)
14676, 145syl 17 . . . . 5 (𝜑 → (∫2𝐺) ∈ ℝ*)
147 supxrleub 12365 . . . . 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)))
14883, 146, 147syl2anc 579 . . . 4 (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧 ≤ (∫2𝐺)))
149144, 148mpbird 248 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) ≤ (∫2𝐺))
15010, 149syl5eqbr 4846 . 2 (𝜑𝑆 ≤ (∫2𝐺))
151 xrletri3 12194 . . 3 (((∫2𝐺) ∈ ℝ*𝑆 ∈ ℝ*) → ((∫2𝐺) = 𝑆 ↔ ((∫2𝐺) ≤ 𝑆𝑆 ≤ (∫2𝐺))))
152146, 86, 151syl2anc 579 . 2 (𝜑 → ((∫2𝐺) = 𝑆 ↔ ((∫2𝐺) ≤ 𝑆𝑆 ≤ (∫2𝐺))))
15389, 150, 152mpbir2and 704 1 (𝜑 → (∫2𝐺) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  wss 3734  c0 4081   class class class wbr 4811  cmpt 4890  dom cdm 5279  ran crn 5280   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6846  𝑟 cofr 7098  supcsup 8557  cr 10192  0cc0 10193  1c1 10194   + caddc 10196  +∞cpnf 10329  *cxr 10331   < clt 10332  cle 10333  cn 11279  [,)cico 12386  [,]cicc 12387  MblFncmbf 23688  1citg1 23689  2citg2 23690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cc 9514  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-disj 4780  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-ofr 7100  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-omul 7773  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-acn 9023  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10527  df-neg 10528  df-div 10944  df-nn 11280  df-2 11340  df-3 11341  df-n0 11544  df-z 11630  df-uz 11894  df-q 11997  df-rp 12036  df-xneg 12153  df-xadd 12154  df-xmul 12155  df-ioo 12388  df-ioc 12389  df-ico 12390  df-icc 12391  df-fz 12541  df-fzo 12681  df-fl 12808  df-seq 13016  df-exp 13075  df-hash 13329  df-cj 14140  df-re 14141  df-im 14142  df-sqrt 14276  df-abs 14277  df-clim 14520  df-rlim 14521  df-sum 14718  df-rest 16365  df-topgen 16386  df-psmet 20027  df-xmet 20028  df-met 20029  df-bl 20030  df-mopn 20031  df-top 20994  df-topon 21011  df-bases 21046  df-cmp 21486  df-ovol 23538  df-vol 23539  df-mbf 23693  df-itg1 23694  df-itg2 23695
This theorem is referenced by:  itg2i1fseq  23829  itg2cnlem1  23835
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