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Theorem itg2monolem1 25020
Description: Lemma for itg2mono 25023. We show that for any constant 𝑡 less than one, 𝑡 · ∫1𝐻 is less than 𝑆, and so 1𝐻𝑆, which is one half of the equality in itg2mono 25023. Consider the sequence 𝐴(𝑛) = {𝑥𝑡 · 𝐻𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is , and so 𝐻𝐴(𝑛) has an integral which equals 1𝐻 in the limit, by itg1climres 24984. Then by taking the limit in (𝑡 · 𝐻) ↾ 𝐴(𝑛) ≤ 𝐹(𝑛), we get 𝑡 · ∫1𝐻𝑆 as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-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*, < )
itg2mono.7 (𝜑𝑇 ∈ (0(,)1))
itg2mono.8 (𝜑𝐻 ∈ dom ∫1)
itg2mono.9 (𝜑𝐻r𝐺)
itg2mono.10 (𝜑𝑆 ∈ ℝ)
itg2mono.11 𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
Assertion
Ref Expression
itg2monolem1 (𝜑 → (𝑇 · (∫1𝐻)) ≤ 𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑛,𝑦,𝐺   𝑛,𝐻,𝑥,𝑦   𝑛,𝐹,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦   𝑆,𝑛,𝑥,𝑦   𝑇,𝑛,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑛)

Proof of Theorem itg2monolem1
Dummy variables 𝑗 𝑘 𝑚 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 12726 . 2 ℕ = (ℤ‘1)
2 1zzd 12456 . 2 (𝜑 → 1 ∈ ℤ)
3 simpr 486 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
4 readdcl 11059 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
54adantl 483 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
6 itg2mono.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
7 rge0ssre 13293 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ
8 fss 6672 . . . . . . . . . . . . . . . 16 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
96, 7, 8sylancl 587 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
10 itg2mono.8 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻 ∈ dom ∫1)
11 itg2mono.7 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑇 ∈ (0(,)1))
12 0xr 11127 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ*
13 1xr 11139 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ*
14 elioo2 13225 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1)))
1512, 13, 14mp2an 690 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1611, 15sylib 217 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1716simp1d 1142 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ℝ)
1817renegcld 11507 . . . . . . . . . . . . . . . . . 18 (𝜑 → -𝑇 ∈ ℝ)
1910, 18i1fmulc 24973 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ × {-𝑇}) ∘f · 𝐻) ∈ dom ∫1)
2019adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘f · 𝐻) ∈ dom ∫1)
21 i1ff 24945 . . . . . . . . . . . . . . . 16 (((ℝ × {-𝑇}) ∘f · 𝐻) ∈ dom ∫1 → ((ℝ × {-𝑇}) ∘f · 𝐻):ℝ⟶ℝ)
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘f · 𝐻):ℝ⟶ℝ)
23 reex 11067 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2423a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ℝ ∈ V)
25 inidm 4169 . . . . . . . . . . . . . . 15 (ℝ ∩ ℝ) = ℝ
265, 9, 22, 24, 24, 25off 7617 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)):ℝ⟶ℝ)
2726adantr 482 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)):ℝ⟶ℝ)
2827ffnd 6656 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) Fn ℝ)
29 elpreima 6995 . . . . . . . . . . . 12 (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) Fn ℝ → (𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ (𝑥 ∈ ℝ ∧ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0))))
3028, 29syl 17 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ (𝑥 ∈ ℝ ∧ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0))))
313, 30mpbirand 705 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0)))
32 elioomnf 13281 . . . . . . . . . . . 12 (0 ∈ ℝ* → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ ℝ ∧ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0)))
3312, 32ax-mp 5 . . . . . . . . . . 11 ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ ℝ ∧ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0))
3426ffvelcdmda 7021 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ ℝ)
3534biantrurd 534 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0 ↔ ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ ℝ ∧ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0)))
3633, 35bitr4id 290 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0))
376ffnd 6656 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) Fn ℝ)
3822ffnd 6656 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘f · 𝐻) Fn ℝ)
39 eqidd 2738 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
4018adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → -𝑇 ∈ ℝ)
41 i1ff 24945 . . . . . . . . . . . . . . . . . . 19 (𝐻 ∈ dom ∫1𝐻:ℝ⟶ℝ)
4210, 41syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻:ℝ⟶ℝ)
4342ffnd 6656 . . . . . . . . . . . . . . . . 17 (𝜑𝐻 Fn ℝ)
4443adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → 𝐻 Fn ℝ)
45 eqidd 2738 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) = (𝐻𝑥))
4624, 40, 44, 45ofc1 7625 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘f · 𝐻)‘𝑥) = (-𝑇 · (𝐻𝑥)))
4717recnd 11108 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 ∈ ℂ)
4847ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ)
4942ffvelcdmda 7021 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5049adantlr 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5150recnd 11108 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℂ)
5248, 51mulneg1d 11533 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (-𝑇 · (𝐻𝑥)) = -(𝑇 · (𝐻𝑥)))
5346, 52eqtrd 2777 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘f · 𝐻)‘𝑥) = -(𝑇 · (𝐻𝑥)))
5437, 38, 24, 24, 25, 39, 53ofval 7610 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))))
559ffvelcdmda 7021 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
5655recnd 11108 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℂ)
5717adantr 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
5857, 49remulcld 11110 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
5958adantlr 713 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
6059recnd 11108 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℂ)
6156, 60negsubd 11443 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6254, 61eqtrd 2777 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6362breq1d 5106 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0 ↔ (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0))
64 0red 11083 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
6555, 59, 64ltsubaddd 11676 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0 ↔ ((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥)))))
6660addid2d 11281 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 + (𝑇 · (𝐻𝑥))) = (𝑇 · (𝐻𝑥)))
6766breq2d 5108 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥))) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
6863, 65, 673bitrd 305 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0 ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
6931, 36, 683bitrd 305 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7069notbid 318 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
71 eldif 3911 . . . . . . . . . 10 (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))))
7271baib 537 . . . . . . . . 9 (𝑥 ∈ ℝ → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))))
7372adantl 483 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))))
7459, 55lenltd 11226 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7570, 73, 743bitr4d 311 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)))
7675rabbi2dva 4168 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)))) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
77 rembl 24809 . . . . . . 7 ℝ ∈ dom vol
78 itg2mono.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
79 i1fmbf 24944 . . . . . . . . . . 11 (((ℝ × {-𝑇}) ∘f · 𝐻) ∈ dom ∫1 → ((ℝ × {-𝑇}) ∘f · 𝐻) ∈ MblFn)
8020, 79syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘f · 𝐻) ∈ MblFn)
8178, 80mbfadd 24930 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) ∈ MblFn)
82 mbfima 24899 . . . . . . . . 9 ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) ∈ MblFn ∧ ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)):ℝ⟶ℝ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ∈ dom vol)
8381, 26, 82syl2anc 585 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ∈ dom vol)
84 cmmbl 24803 . . . . . . . 8 ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ∈ dom vol → (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ∈ dom vol)
8583, 84syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ∈ dom vol)
86 inmbl 24811 . . . . . . 7 ((ℝ ∈ dom vol ∧ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ∈ dom vol) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)))) ∈ dom vol)
8777, 85, 86sylancr 588 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)))) ∈ dom vol)
8876, 87eqeltrrd 2839 . . . . 5 ((𝜑𝑛 ∈ ℕ) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} ∈ dom vol)
89 itg2mono.11 . . . . 5 𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
9088, 89fmptd 7048 . . . 4 (𝜑𝐴:ℕ⟶dom vol)
91 itg2mono.4 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
9291ralrimiva 3140 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
93 fveq2 6829 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
94 fvoveq1 7364 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑗 + 1)))
9593, 94breq12d 5109 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)) ↔ (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1))))
9695cbvralvw 3222 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)))
9792, 96sylib 217 . . . . . . . . . 10 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)))
9897r19.21bi 3231 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)))
996ralrimiva 3140 . . . . . . . . . . . . 13 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
10093feq1d 6640 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑗):ℝ⟶(0[,)+∞)))
101100cbvralvw 3222 . . . . . . . . . . . . 13 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
10299, 101sylib 217 . . . . . . . . . . . 12 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
103102r19.21bi 3231 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
104103ffnd 6656 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) Fn ℝ)
105 peano2nn 12090 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
106 fveq2 6829 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (𝐹𝑛) = (𝐹‘(𝑗 + 1)))
107106feq1d 6640 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞)))
108107rspccva 3572 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (𝑗 + 1) ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
10999, 105, 108syl2an 597 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
110109ffnd 6656 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) Fn ℝ)
11123a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ℝ ∈ V)
112 eqidd 2738 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑥))
113 eqidd 2738 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
114104, 110, 111, 111, 25, 112, 113ofrfval 7609 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)) ↔ ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
11598, 114mpbid 231 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
116115r19.21bi 3231 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
11717ad2antrr 724 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
11842adantr 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 𝐻:ℝ⟶ℝ)
119118ffvelcdmda 7021 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
120117, 119remulcld 11110 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
121 fss 6672 . . . . . . . . . 10 (((𝐹𝑗):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑗):ℝ⟶ℝ)
122103, 7, 121sylancl 587 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶ℝ)
123122ffvelcdmda 7021 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
124 fss 6672 . . . . . . . . . 10 (((𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
125109, 7, 124sylancl 587 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
126125ffvelcdmda 7021 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) ∈ ℝ)
127 letr 11174 . . . . . . . 8 (((𝑇 · (𝐻𝑥)) ∈ ℝ ∧ ((𝐹𝑗)‘𝑥) ∈ ℝ ∧ ((𝐹‘(𝑗 + 1))‘𝑥) ∈ ℝ) → (((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) ∧ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
128120, 123, 126, 127syl3anc 1371 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) ∧ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
129116, 128mpan2d 692 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
130129ss2rabdv 4024 . . . . 5 ((𝜑𝑗 ∈ ℕ) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ⊆ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
13193fveq1d 6831 . . . . . . . . 9 (𝑛 = 𝑗 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑗)‘𝑥))
132131breq2d 5108 . . . . . . . 8 (𝑛 = 𝑗 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
133132rabbidv 3412 . . . . . . 7 (𝑛 = 𝑗 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
13423rabex 5280 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ∈ V
135133, 89, 134fvmpt 6935 . . . . . 6 (𝑗 ∈ ℕ → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
136135adantl 483 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
137105adantl 483 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
138106fveq1d 6831 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐹𝑛)‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
139138breq2d 5108 . . . . . . . 8 (𝑛 = (𝑗 + 1) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
140139rabbidv 3412 . . . . . . 7 (𝑛 = (𝑗 + 1) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
14123rabex 5280 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)} ∈ V
142140, 89, 141fvmpt 6935 . . . . . 6 ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
143137, 142syl 17 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐴‘(𝑗 + 1)) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
144130, 136, 1433sstr4d 3982 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ⊆ (𝐴‘(𝑗 + 1)))
14558adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
14649adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℝ)
14755an32s 650 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
148147fmpttd 7049 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
149148frnd 6663 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
150 1nn 12089 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
151 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
152151, 147dmmptd 6633 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
153150, 152eleqtrrid 2845 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
154153ne0d 4286 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
155 dm0rn0 5870 . . . . . . . . . . . . . . . . . 18 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
156155necon3bii 2994 . . . . . . . . . . . . . . . . 17 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
157154, 156sylib 217 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
158 itg2mono.5 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
159148ffnd 6656 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
160 breq1 5099 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
161160ralrn 7024 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
162159, 161syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
163 fveq2 6829 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
164163fveq1d 6831 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
165 fvex 6842 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑚)‘𝑥) ∈ V
166164, 151, 165fvmpt 6935 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
167166breq1d 5106 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
168167ralbiia 3091 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
169164breq1d 5106 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
170169cbvralvw 3222 . . . . . . . . . . . . . . . . . . . 20 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
171168, 170bitr4i 278 . . . . . . . . . . . . . . . . . . 19 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
172162, 171bitrdi 287 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
173172rexbidv 3172 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
174158, 173mpbird 257 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
175149, 157, 174suprcld 12043 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
176175adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
17716simp3d 1144 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 < 1)
178177adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 < 1)
17917adantr 482 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 ∈ ℝ)
180 1red 11081 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 1 ∈ ℝ)
181 simprr 771 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 0 < (𝐻𝑥))
182 ltmul1 11930 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝐻𝑥) ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
183179, 180, 146, 181, 182syl112anc 1374 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
184178, 183mpbid 231 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥)))
185146recnd 11108 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℂ)
186185mulid2d 11098 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (1 · (𝐻𝑥)) = (𝐻𝑥))
187184, 186breqtrd 5122 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < (𝐻𝑥))
188 itg2mono.9 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻r𝐺)
189 itg2mono.1 . . . . . . . . . . . . . . . . . . . . 21 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
190175, 189fmptd 7048 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:ℝ⟶ℝ)
191190ffnd 6656 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 Fn ℝ)
19223a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ℝ ∈ V)
193 eqidd 2738 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐻𝑦) = (𝐻𝑦))
194 fveq2 6829 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑦 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑦))
195194mpteq2dv 5198 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
196195rneqd 5883 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
197196supeq1d 9307 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
198 ltso 11160 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
199198supex 9324 . . . . . . . . . . . . . . . . . . . . 21 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ) ∈ V
200197, 189, 199fvmpt 6935 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℝ → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
201200adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
20243, 191, 192, 192, 25, 193, 201ofrfval 7609 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐻r𝐺 ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
203188, 202mpbid 231 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
204 fveq2 6829 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐻𝑥) = (𝐻𝑦))
205204, 197breq12d 5109 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
206205cbvralvw 3222 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
207203, 206sylibr 233 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
208207r19.21bi 3231 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
209208adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
210145, 146, 176, 187, 209ltletrd 11240 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
211149adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
212157adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
213174adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
214 suprlub 12044 . . . . . . . . . . . . . 14 (((ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦) ∧ (𝑇 · (𝐻𝑥)) ∈ ℝ) → ((𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤))
215211, 212, 213, 145, 214syl31anc 1373 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ((𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤))
216210, 215mpbid 231 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤)
217159adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
218 breq2 5100 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) → ((𝑇 · (𝐻𝑥)) < 𝑤 ↔ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗)))
219218rexrn 7023 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤 ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗)))
220217, 219syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤 ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗)))
221 fvex 6842 . . . . . . . . . . . . . . . 16 ((𝐹𝑗)‘𝑥) ∈ V
222131, 151, 221fvmpt 6935 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) = ((𝐹𝑗)‘𝑥))
223222breq2d 5108 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ((𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
224223rexbiia 3092 . . . . . . . . . . . . 13 (∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
225220, 224bitrdi 287 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤 ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
226216, 225mpbid 231 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
227179, 146remulcld 11110 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
228103adantlr 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
229 simplr 767 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑥 ∈ ℝ)
230228, 229ffvelcdmd 7022 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ (0[,)+∞))
231 elrege0 13291 . . . . . . . . . . . . . . . 16 (((𝐹𝑗)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
232230, 231sylib 217 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
233232simpld 496 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
234233adantlrr 719 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
235 ltle 11168 . . . . . . . . . . . . 13 (((𝑇 · (𝐻𝑥)) ∈ ℝ ∧ ((𝐹𝑗)‘𝑥) ∈ ℝ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
236227, 234, 235syl2an2r 683 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
237236reximdva 3162 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
238226, 237mpd 15 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
239238anassrs 469 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 0 < (𝐻𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
240150ne0ii 4288 . . . . . . . . . . 11 ℕ ≠ ∅
24158adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
242241adantr 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
243 0red 11083 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ)
244232adantlrr 719 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
245244simpld 496 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
246 simplrr 776 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ≤ 0)
24749adantrr 715 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝐻𝑥) ∈ ℝ)
248247adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ∈ ℝ)
24917ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
25016simp2d 1143 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑇)
251250ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 < 𝑇)
252 lemul2 11933 . . . . . . . . . . . . . . . 16 (((𝐻𝑥) ∈ ℝ ∧ 0 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
253248, 243, 249, 251, 252syl112anc 1374 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
254246, 253mpbid 231 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0))
255249recnd 11108 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℂ)
256255mul01d 11279 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · 0) = 0)
257254, 256breqtrd 5122 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ 0)
258244simprd 497 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ≤ ((𝐹𝑗)‘𝑥))
259242, 243, 245, 257, 258letrd 11237 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
260259ralrimiva 3140 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
261 r19.2z 4443 . . . . . . . . . . 11 ((ℕ ≠ ∅ ∧ ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
262240, 260, 261sylancr 588 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
263262anassrs 469 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝐻𝑥) ≤ 0) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
264 0red 11083 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
265239, 263, 264, 49ltlecasei 11188 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
266265ralrimiva 3140 . . . . . . 7 (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
267 rabid2 3433 . . . . . . 7 (ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
268266, 267sylibr 233 . . . . . 6 (𝜑 → ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
269 iunrab 5003 . . . . . 6 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)}
270268, 269eqtr4di 2795 . . . . 5 (𝜑 → ℝ = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
271136iuneq2dv 4969 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
27290ffnd 6656 . . . . . 6 (𝜑𝐴 Fn ℕ)
273 fniunfv 7180 . . . . . 6 (𝐴 Fn ℕ → 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
274272, 273syl 17 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
275270, 271, 2743eqtr2rd 2784 . . . 4 (𝜑 ran 𝐴 = ℝ)
276 eqid 2737 . . . 4 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))
27790, 144, 275, 10, 276itg1climres 24984 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ⇝ (∫1𝐻))
278 nnex 12084 . . . . 5 ℕ ∈ V
279278mptex 7159 . . . 4 (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ∈ V
280279a1i 11 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ∈ V)
281 fveq2 6829 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
282281eleq2d 2823 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑥 ∈ (𝐴𝑗) ↔ 𝑥 ∈ (𝐴𝑘)))
283282ifbid 4500 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0) = if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
284283mpteq2dv 5198 . . . . . . . 8 (𝑗 = 𝑘 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
285284fveq2d 6833 . . . . . . 7 (𝑗 = 𝑘 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
286 eqid 2737 . . . . . . 7 (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))
287 fvex 6842 . . . . . . 7 (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ V
288285, 286, 287fvmpt 6935 . . . . . 6 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
289288adantl 483 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
29090ffvelcdmda 7021 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐴𝑘) ∈ dom vol)
291 eqid 2737 . . . . . . . 8 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
292291i1fres 24975 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑘) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
29310, 290, 292syl2an2r 683 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
294 itg1cl 24954 . . . . . 6 ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ ℝ)
295293, 294syl 17 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ ℝ)
296289, 295eqeltrd 2838 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℝ)
297296recnd 11108 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℂ)
298285oveq2d 7357 . . . . . 6 (𝑗 = 𝑘 → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
299 eqid 2737 . . . . . 6 (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) = (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))
300 ovex 7374 . . . . . 6 (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) ∈ V
301298, 299, 300fvmpt 6935 . . . . 5 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
302288oveq2d 7357 . . . . 5 (𝑘 ∈ ℕ → (𝑇 · ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘)) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
303301, 302eqtr4d 2780 . . . 4 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘)))
304303adantl 483 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘)))
3051, 2, 277, 47, 280, 297, 304climmulc2 15445 . 2 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ⇝ (𝑇 · (∫1𝐻)))
306 icossicc 13273 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
307 fss 6672 . . . . . . 7 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
3086, 306, 307sylancl 587 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
309 itg2mono.10 . . . . . . 7 (𝜑𝑆 ∈ ℝ)
310309adantr 482 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
311 itg2cl 25002 . . . . . . . . . . 11 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
312308, 311syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
313312fmpttd 7049 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
314313frnd 6663 . . . . . . . 8 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
315 fvex 6842 . . . . . . . . . . 11 (∫2‘(𝐹𝑛)) ∈ V
316315elabrex 7176 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
317316adantl 483 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
318 eqid 2737 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
319318rnmpt 5900 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))}
320317, 319eleqtrrdi 2849 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))))
321 supxrub 13163 . . . . . . . 8 ((ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ* ∧ (∫2‘(𝐹𝑛)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))) → (∫2‘(𝐹𝑛)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ))
322314, 320, 321syl2an2r 683 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ))
323 itg2mono.6 . . . . . . 7 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )
324322, 323breqtrrdi 5138 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ 𝑆)
325 itg2lecl 25008 . . . . . 6 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ ∧ (∫2‘(𝐹𝑛)) ≤ 𝑆) → (∫2‘(𝐹𝑛)) ∈ ℝ)
326308, 310, 324, 325syl3anc 1371 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ)
327326fmpttd 7049 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ)
328308ralrimiva 3140 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞))
329 fveq2 6829 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
330329feq1d 6640 . . . . . . . . . . 11 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹𝑘):ℝ⟶(0[,]+∞)))
331330cbvralvw 3222 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
332328, 331sylib 217 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
333 peano2nn 12090 . . . . . . . . 9 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
334 fveq2 6829 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
335334feq1d 6640 . . . . . . . . . 10 (𝑘 = (𝑛 + 1) → ((𝐹𝑘):ℝ⟶(0[,]+∞) ↔ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞)))
336335rspccva 3572 . . . . . . . . 9 ((∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞) ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
337332, 333, 336syl2an 597 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
338 itg2le 25009 . . . . . . . 8 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞) ∧ (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
339308, 337, 91, 338syl3anc 1371 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
340339ralrimiva 3140 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
341 2fveq3 6834 . . . . . . . . . 10 (𝑛 = 𝑘 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑘)))
342 fvex 6842 . . . . . . . . . 10 (∫2‘(𝐹𝑘)) ∈ V
343341, 318, 342fvmpt 6935 . . . . . . . . 9 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
344 peano2nn 12090 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
345 2fveq3 6834 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹‘(𝑘 + 1))))
346 fvex 6842 . . . . . . . . . . 11 (∫2‘(𝐹‘(𝑘 + 1))) ∈ V
347345, 318, 346fvmpt 6935 . . . . . . . . . 10 ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) = (∫2‘(𝐹‘(𝑘 + 1))))
348344, 347syl 17 . . . . . . . . 9 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) = (∫2‘(𝐹‘(𝑘 + 1))))
349343, 348breq12d 5109 . . . . . . . 8 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
350349ralbiia 3091 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
351 fvoveq1 7364 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
352351fveq2d 6833 . . . . . . . . 9 (𝑛 = 𝑘 → (∫2‘(𝐹‘(𝑛 + 1))) = (∫2‘(𝐹‘(𝑘 + 1))))
353341, 352breq12d 5109 . . . . . . . 8 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
354353cbvralvw 3222 . . . . . . 7 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
355350, 354bitr4i 278 . . . . . 6 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
356340, 355sylibr 233 . . . . 5 (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
357356r19.21bi 3231 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
358324ralrimiva 3140 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆)
359343breq1d 5106 . . . . . . . . 9 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
360359ralbiia 3091 . . . . . . . 8 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
361341breq1d 5106 . . . . . . . . 9 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
362361cbvralvw 3222 . . . . . . . 8 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
363360, 362bitr4i 278 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥)
364 breq2 5100 . . . . . . . 8 (𝑥 = 𝑆 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑛)) ≤ 𝑆))
365364ralbidv 3171 . . . . . . 7 (𝑥 = 𝑆 → (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
366363, 365bitrid 283 . . . . . 6 (𝑥 = 𝑆 → (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
367366rspcev 3573 . . . . 5 ((𝑆 ∈ ℝ ∧ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
368309, 358, 367syl2anc 585 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
3691, 2, 327, 357, 368climsup 15480 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
370327frnd 6663 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ)
371318, 312dmmptd 6633 . . . . . . 7 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ℕ)
372240a1i 11 . . . . . . 7 (𝜑 → ℕ ≠ ∅)
373371, 372eqnetrd 3009 . . . . . 6 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
374 dm0rn0 5870 . . . . . . 7 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅)
375374necon3bii 2994 . . . . . 6 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
376373, 375sylib 217 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
377315, 318fnmpti 6631 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ
378 breq1 5099 . . . . . . . . 9 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) → (𝑧𝑥 ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
379378ralrn 7024 . . . . . . . 8 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
380377, 379mp1i 13 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
381380rexbidv 3172 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
382368, 381mpbird 257 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥)
383 supxrre 13166 . . . . 5 ((ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥) → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
384370, 376, 382, 383syl3anc 1371 . . . 4 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
385323, 384eqtr2id 2790 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ) = 𝑆)
386369, 385breqtrd 5122 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ 𝑆)
38717adantr 482 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
38890ffvelcdmda 7021 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ∈ dom vol)
389276i1fres 24975 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑗) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
39010, 388, 389syl2an2r 683 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
391 itg1cl 24954 . . . . . 6 ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) ∈ ℝ)
392390, 391syl 17 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) ∈ ℝ)
393387, 392remulcld 11110 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ∈ ℝ)
394393fmpttd 7049 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))):ℕ⟶ℝ)
395394ffvelcdmda 7021 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) ∈ ℝ)
396327ffvelcdmda 7021 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ∈ ℝ)
397329feq1d 6640 . . . . . . . 8 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑘):ℝ⟶(0[,)+∞)))
398397cbvralvw 3222 . . . . . . 7 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
39999, 398sylib 217 . . . . . 6 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
400399r19.21bi 3231 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,)+∞))
401 fss 6672 . . . . 5 (((𝐹𝑘):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑘):ℝ⟶(0[,]+∞))
402400, 306, 401sylancl 587 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,]+∞))
40323a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ℝ ∈ V)
40417adantr 482 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℝ)
405404adantr 482 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
406 fvex 6842 . . . . . . . . 9 (𝐻𝑥) ∈ V
407 c0ex 11074 . . . . . . . . 9 0 ∈ V
408406, 407ifex 4527 . . . . . . . 8 if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V
409408a1i 11 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V)
410 fconstmpt 5684 . . . . . . . 8 (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇)
411410a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇))
412 eqidd 2738 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
413403, 405, 409, 411, 412offval2 7619 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
414 ovif2 7439 . . . . . . . 8 (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0))
41547adantr 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℂ)
416415mul01d 11279 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝑇 · 0) = 0)
417416ifeq2d 4497 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
418414, 417eqtrid 2789 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
419418mpteq2dv 5198 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
420413, 419eqtrd 2777 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
421293, 404i1fmulc 24973 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ dom ∫1)
422420, 421eqeltrrd 2839 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∈ dom ∫1)
423 iftrue 4483 . . . . . . . . 9 (𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
424423adantl 483 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
425329fveq1d 6831 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑘)‘𝑥))
426425breq2d 5108 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
427426rabbidv 3412 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
42823rabex 5280 . . . . . . . . . . . . 13 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ∈ V
429427, 89, 428fvmpt 6935 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
430429ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
431430eleq2d 2823 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐴𝑘) ↔ 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)}))
432431biimpa 478 . . . . . . . . 9 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
433 rabid 3424 . . . . . . . . . 10 (𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ↔ (𝑥 ∈ ℝ ∧ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
434433simprbi 498 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥))
435432, 434syl 17 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥))
436424, 435eqbrtrd 5118 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
437 iffalse 4486 . . . . . . . . 9 𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
438437adantl 483 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
439400ffvelcdmda 7021 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ (0[,)+∞))
440 elrege0 13291 . . . . . . . . . . 11 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑘)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑘)‘𝑥)))
441440simprbi 498 . . . . . . . . . 10 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) → 0 ≤ ((𝐹𝑘)‘𝑥))
442439, 441syl 17 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹𝑘)‘𝑥))
443442adantr 482 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → 0 ≤ ((𝐹𝑘)‘𝑥))
444438, 443eqbrtrd 5118 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
445436, 444pm2.61dan 811 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
446445ralrimiva 3140 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
447 ovex 7374 . . . . . . . 8 (𝑇 · (𝐻𝑥)) ∈ V
448447, 407ifex 4527 . . . . . . 7 if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V
449448a1i 11 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V)
450 fvexd 6844 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ V)
451 eqidd 2738 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
452400feqmptd 6897 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = (𝑥 ∈ ℝ ↦ ((𝐹𝑘)‘𝑥)))
453403, 449, 450, 451, 452ofrfval2 7620 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘r ≤ (𝐹𝑘) ↔ ∀𝑥 ∈ ℝ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥)))
454446, 453mpbird 257 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘r ≤ (𝐹𝑘))
455 itg2ub 25003 . . . 4 (((𝐹𝑘):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∈ dom ∫1 ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘r ≤ (𝐹𝑘)) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))) ≤ (∫2‘(𝐹𝑘)))
456402, 422, 454, 455syl3anc 1371 . . 3 ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))) ≤ (∫2‘(𝐹𝑘)))
457301adantl 483 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
458293, 404itg1mulc 24974 . . . 4 ((𝜑𝑘 ∈ ℕ) → (∫1‘((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
459420fveq2d 6833 . . . 4 ((𝜑𝑘 ∈ ℕ) → (∫1‘((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))))
460457, 458, 4593eqtr2d 2783 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))))
461343adantl 483 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
462456, 460, 4613brtr4d 5128 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘))
4631, 2, 305, 386, 395, 396, 462climle 15448 1 (𝜑 → (𝑇 · (∫1𝐻)) ≤ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  {cab 2714  wne 2941  wral 3062  wrex 3071  {crab 3404  Vcvv 3442  cdif 3898  cin 3900  wss 3901  c0 4273  ifcif 4477  {csn 4577   cuni 4856   ciun 4945   class class class wbr 5096  cmpt 5179   × cxp 5622  ccnv 5623  dom cdm 5624  ran crn 5625  cima 5627   Fn wfn 6478  wf 6479  cfv 6483  (class class class)co 7341  f cof 7597  r cofr 7598  supcsup 9301  cc 10974  cr 10975  0cc0 10976  1c1 10977   + caddc 10979   · cmul 10981  +∞cpnf 11111  -∞cmnf 11112  *cxr 11113   < clt 11114  cle 11115  cmin 11310  -cneg 11311  cn 12078  (,)cioo 13184  [,)cico 13186  [,]cicc 13187  cli 15292  volcvol 24732  MblFncmbf 24883  1citg1 24884  2citg2 24885
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-inf2 9502  ax-cc 10296  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053  ax-pre-sup 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-int 4899  df-iun 4947  df-disj 5062  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-se 5580  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-isom 6492  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7599  df-ofr 7600  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-2o 8372  df-oadd 8375  df-omul 8376  df-er 8573  df-map 8692  df-pm 8693  df-en 8809  df-dom 8810  df-sdom 8811  df-fin 8812  df-fi 9272  df-sup 9303  df-inf 9304  df-oi 9371  df-dju 9762  df-card 9800  df-acn 9803  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-div 11738  df-nn 12079  df-2 12141  df-3 12142  df-n0 12339  df-z 12425  df-uz 12688  df-q 12794  df-rp 12836  df-xneg 12953  df-xadd 12954  df-xmul 12955  df-ioo 13188  df-ioc 13189  df-ico 13190  df-icc 13191  df-fz 13345  df-fzo 13488  df-fl 13617  df-seq 13827  df-exp 13888  df-hash 14150  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-clim 15296  df-rlim 15297  df-sum 15497  df-rest 17230  df-topgen 17251  df-psmet 20694  df-xmet 20695  df-met 20696  df-bl 20697  df-mopn 20698  df-top 22148  df-topon 22165  df-bases 22201  df-cmp 22643  df-ovol 24733  df-vol 24734  df-mbf 24888  df-itg1 24889  df-itg2 24890
This theorem is referenced by:  itg2monolem3  25022
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