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Theorem itg2monolem1 24357
Description: Lemma for itg2mono 24360. 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 24360. 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 24321. 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 12278 . 2 ℕ = (ℤ‘1)
2 1zzd 12010 . 2 (𝜑 → 1 ∈ ℤ)
3 simpr 488 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
4 readdcl 10618 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
54adantl 485 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
6 itg2mono.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
7 rge0ssre 12843 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ
8 fss 6517 . . . . . . . . . . . . . . . 16 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
96, 7, 8sylancl 589 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
10 itg2mono.8 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻 ∈ dom ∫1)
11 itg2mono.7 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑇 ∈ (0(,)1))
12 0xr 10686 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ*
13 1xr 10698 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ*
14 elioo2 12776 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1)))
1512, 13, 14mp2an 691 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1611, 15sylib 221 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1716simp1d 1139 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ℝ)
1817renegcld 11065 . . . . . . . . . . . . . . . . . 18 (𝜑 → -𝑇 ∈ ℝ)
1910, 18i1fmulc 24310 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ × {-𝑇}) ∘f · 𝐻) ∈ dom ∫1)
2019adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘f · 𝐻) ∈ dom ∫1)
21 i1ff 24283 . . . . . . . . . . . . . . . 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 10626 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2423a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ℝ ∈ V)
25 inidm 4180 . . . . . . . . . . . . . . 15 (ℝ ∩ ℝ) = ℝ
265, 9, 22, 24, 24, 25off 7418 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)):ℝ⟶ℝ)
2726adantr 484 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)):ℝ⟶ℝ)
2827ffnd 6504 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) Fn ℝ)
29 elpreima 6819 . . . . . . . . . . . 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 706 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0)))
3226ffvelrnda 6842 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ ℝ)
3332biantrurd 536 . . . . . . . . . . 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)))
34 elioomnf 12831 . . . . . . . . . . . 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)))
3512, 34ax-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))
3633, 35syl6rbbr 293 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0))
376ffnd 6504 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) Fn ℝ)
3822ffnd 6504 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘f · 𝐻) Fn ℝ)
39 eqidd 2825 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
4018adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → -𝑇 ∈ ℝ)
41 i1ff 24283 . . . . . . . . . . . . . . . . . . 19 (𝐻 ∈ dom ∫1𝐻:ℝ⟶ℝ)
4210, 41syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻:ℝ⟶ℝ)
4342ffnd 6504 . . . . . . . . . . . . . . . . 17 (𝜑𝐻 Fn ℝ)
4443adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → 𝐻 Fn ℝ)
45 eqidd 2825 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) = (𝐻𝑥))
4624, 40, 44, 45ofc1 7426 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘f · 𝐻)‘𝑥) = (-𝑇 · (𝐻𝑥)))
4717recnd 10667 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 ∈ ℂ)
4847ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ)
4942ffvelrnda 6842 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5049adantlr 714 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5150recnd 10667 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℂ)
5248, 51mulneg1d 11091 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (-𝑇 · (𝐻𝑥)) = -(𝑇 · (𝐻𝑥)))
5346, 52eqtrd 2859 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘f · 𝐻)‘𝑥) = -(𝑇 · (𝐻𝑥)))
5437, 38, 24, 24, 25, 39, 53ofval 7412 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))))
559ffvelrnda 6842 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
5655recnd 10667 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℂ)
5717adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
5857, 49remulcld 10669 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
5958adantlr 714 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
6059recnd 10667 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℂ)
6156, 60negsubd 11001 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6254, 61eqtrd 2859 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6362breq1d 5062 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0 ↔ (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0))
64 0red 10642 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
6555, 59, 64ltsubaddd 11234 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0 ↔ ((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥)))))
6660addid2d 10839 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 + (𝑇 · (𝐻𝑥))) = (𝑇 · (𝐻𝑥)))
6766breq2d 5064 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥))) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
6863, 65, 673bitrd 308 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻))‘𝑥) < 0 ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
6931, 36, 683bitrd 308 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7069notbid 321 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
71 eldif 3929 . . . . . . . . . 10 (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))))
7271baib 539 . . . . . . . . 9 (𝑥 ∈ ℝ → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))))
7372adantl 485 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))))
7459, 55lenltd 10784 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7570, 73, 743bitr4d 314 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0))) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)))
7675rabbi2dva 4179 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)))) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
77 rembl 24147 . . . . . . 7 ℝ ∈ dom vol
78 itg2mono.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
79 i1fmbf 24282 . . . . . . . . . . 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 24268 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) ∈ MblFn)
82 mbfima 24237 . . . . . . . . 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 587 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)) ∈ dom vol)
84 cmmbl 24141 . . . . . . . 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 24149 . . . . . . 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 590 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘f + ((ℝ × {-𝑇}) ∘f · 𝐻)) “ (-∞(,)0)))) ∈ dom vol)
8876, 87eqeltrrd 2917 . . . . 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 6869 . . . 4 (𝜑𝐴:ℕ⟶dom vol)
91 itg2mono.4 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
9291ralrimiva 3177 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
93 fveq2 6661 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
94 fvoveq1 7172 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑗 + 1)))
9593, 94breq12d 5065 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)) ↔ (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1))))
9695cbvralvw 3434 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)))
9792, 96sylib 221 . . . . . . . . . 10 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)))
9897r19.21bi 3203 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)))
996ralrimiva 3177 . . . . . . . . . . . . 13 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
10093feq1d 6488 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑗):ℝ⟶(0[,)+∞)))
101100cbvralvw 3434 . . . . . . . . . . . . 13 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
10299, 101sylib 221 . . . . . . . . . . . 12 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
103102r19.21bi 3203 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
104103ffnd 6504 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) Fn ℝ)
105 peano2nn 11646 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
106 fveq2 6661 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (𝐹𝑛) = (𝐹‘(𝑗 + 1)))
107106feq1d 6488 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞)))
108107rspccva 3608 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (𝑗 + 1) ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
10999, 105, 108syl2an 598 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
110109ffnd 6504 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) Fn ℝ)
11123a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ℝ ∈ V)
112 eqidd 2825 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑥))
113 eqidd 2825 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
114104, 110, 111, 111, 25, 112, 113ofrfval 7411 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((𝐹𝑗) ∘r ≤ (𝐹‘(𝑗 + 1)) ↔ ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
11598, 114mpbid 235 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
116115r19.21bi 3203 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
11717ad2antrr 725 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
11842adantr 484 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 𝐻:ℝ⟶ℝ)
119118ffvelrnda 6842 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
120117, 119remulcld 10669 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
121 fss 6517 . . . . . . . . . 10 (((𝐹𝑗):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑗):ℝ⟶ℝ)
122103, 7, 121sylancl 589 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶ℝ)
123122ffvelrnda 6842 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
124 fss 6517 . . . . . . . . . 10 (((𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
125109, 7, 124sylancl 589 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
126125ffvelrnda 6842 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) ∈ ℝ)
127 letr 10732 . . . . . . . 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 1368 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) ∧ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
129116, 128mpan2d 693 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
130129ss2rabdv 4038 . . . . 5 ((𝜑𝑗 ∈ ℕ) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ⊆ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
13193fveq1d 6663 . . . . . . . . 9 (𝑛 = 𝑗 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑗)‘𝑥))
132131breq2d 5064 . . . . . . . 8 (𝑛 = 𝑗 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
133132rabbidv 3465 . . . . . . 7 (𝑛 = 𝑗 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
13423rabex 5221 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ∈ V
135133, 89, 134fvmpt 6759 . . . . . 6 (𝑗 ∈ ℕ → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
136135adantl 485 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
137105adantl 485 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
138106fveq1d 6663 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐹𝑛)‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
139138breq2d 5064 . . . . . . . 8 (𝑛 = (𝑗 + 1) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
140139rabbidv 3465 . . . . . . 7 (𝑛 = (𝑗 + 1) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
14123rabex 5221 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)} ∈ V
142140, 89, 141fvmpt 6759 . . . . . 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 4000 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ⊆ (𝐴‘(𝑗 + 1)))
14558adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
14649adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℝ)
14755an32s 651 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
148147fmpttd 6870 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
149148frnd 6510 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
150 1nn 11645 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
151 eqid 2824 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
152151, 147dmmptd 6482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
153150, 152eleqtrrid 2923 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
154153ne0d 4284 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
155 dm0rn0 5782 . . . . . . . . . . . . . . . . . 18 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
156155necon3bii 3066 . . . . . . . . . . . . . . . . 17 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
157154, 156sylib 221 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
158 itg2mono.5 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
159148ffnd 6504 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
160 breq1 5055 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
161160ralrn 6845 . . . . . . . . . . . . . . . . . . . 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 6661 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
164163fveq1d 6663 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
165 fvex 6674 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑚)‘𝑥) ∈ V
166164, 151, 165fvmpt 6759 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
167166breq1d 5062 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
168167ralbiia 3159 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
169164breq1d 5062 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
170169cbvralvw 3434 . . . . . . . . . . . . . . . . . . . 20 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
171168, 170bitr4i 281 . . . . . . . . . . . . . . . . . . 19 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
172162, 171syl6bb 290 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
173172rexbidv 3289 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
174158, 173mpbird 260 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
175149, 157, 174suprcld 11600 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
176175adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
17716simp3d 1141 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 < 1)
178177adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 < 1)
17917adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 ∈ ℝ)
180 1red 10640 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 1 ∈ ℝ)
181 simprr 772 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 0 < (𝐻𝑥))
182 ltmul1 11488 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝐻𝑥) ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
183179, 180, 146, 181, 182syl112anc 1371 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
184178, 183mpbid 235 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥)))
185146recnd 10667 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℂ)
186185mulid2d 10657 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (1 · (𝐻𝑥)) = (𝐻𝑥))
187184, 186breqtrd 5078 . . . . . . . . . . . . . 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 6869 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:ℝ⟶ℝ)
191190ffnd 6504 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 Fn ℝ)
19223a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ℝ ∈ V)
193 eqidd 2825 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐻𝑦) = (𝐻𝑦))
194 fveq2 6661 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑦 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑦))
195194mpteq2dv 5148 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
196195rneqd 5795 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
197196supeq1d 8907 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
198 ltso 10719 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
199198supex 8924 . . . . . . . . . . . . . . . . . . . . 21 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ) ∈ V
200197, 189, 199fvmpt 6759 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℝ → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
201200adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
20243, 191, 192, 192, 25, 193, 201ofrfval 7411 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐻r𝐺 ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
203188, 202mpbid 235 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
204 fveq2 6661 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐻𝑥) = (𝐻𝑦))
205204, 197breq12d 5065 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
206205cbvralvw 3434 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
207203, 206sylibr 237 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
208207r19.21bi 3203 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
209208adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
210145, 146, 176, 187, 209ltletrd 10798 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
211149adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
212157adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
213174adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
214 suprlub 11601 . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤)
217159adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
218 breq2 5056 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) → ((𝑇 · (𝐻𝑥)) < 𝑤 ↔ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗)))
219218rexrn 6844 . . . . . . . . . . . . . 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 6674 . . . . . . . . . . . . . . . 16 ((𝐹𝑗)‘𝑥) ∈ V
222131, 151, 221fvmpt 6759 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) = ((𝐹𝑗)‘𝑥))
223222breq2d 5064 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ((𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
224223rexbiia 3240 . . . . . . . . . . . . 13 (∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
225220, 224syl6bb 290 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤 ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
226216, 225mpbid 235 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
227179, 146remulcld 10669 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
228103adantlr 714 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
229 simplr 768 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑥 ∈ ℝ)
230228, 229ffvelrnd 6843 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ (0[,)+∞))
231 elrege0 12841 . . . . . . . . . . . . . . . 16 (((𝐹𝑗)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
232230, 231sylib 221 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
233232simpld 498 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
234233adantlrr 720 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
235 ltle 10727 . . . . . . . . . . . . 13 (((𝑇 · (𝐻𝑥)) ∈ ℝ ∧ ((𝐹𝑗)‘𝑥) ∈ ℝ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
236227, 234, 235syl2an2r 684 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
237236reximdva 3266 . . . . . . . . . . 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 471 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 0 < (𝐻𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
240150ne0ii 4286 . . . . . . . . . . 11 ℕ ≠ ∅
24158adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
242241adantr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
243 0red 10642 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ)
244232adantlrr 720 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
245244simpld 498 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
246 simplrr 777 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ≤ 0)
24749adantrr 716 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝐻𝑥) ∈ ℝ)
248247adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ∈ ℝ)
24917ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
25016simp2d 1140 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑇)
251250ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 < 𝑇)
252 lemul2 11491 . . . . . . . . . . . . . . . 16 (((𝐻𝑥) ∈ ℝ ∧ 0 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
253248, 243, 249, 251, 252syl112anc 1371 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
254246, 253mpbid 235 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0))
255249recnd 10667 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℂ)
256255mul01d 10837 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · 0) = 0)
257254, 256breqtrd 5078 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ 0)
258244simprd 499 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ≤ ((𝐹𝑗)‘𝑥))
259242, 243, 245, 257, 258letrd 10795 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
260259ralrimiva 3177 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
261 r19.2z 4423 . . . . . . . . . . 11 ((ℕ ≠ ∅ ∧ ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
262240, 260, 261sylancr 590 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
263262anassrs 471 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝐻𝑥) ≤ 0) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
264 0red 10642 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
265239, 263, 264, 49ltlecasei 10746 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
266265ralrimiva 3177 . . . . . . 7 (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
267 rabid2 3372 . . . . . . 7 (ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
268266, 267sylibr 237 . . . . . 6 (𝜑 → ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
269 iunrab 4962 . . . . . 6 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)}
270268, 269syl6eqr 2877 . . . . 5 (𝜑 → ℝ = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
271136iuneq2dv 4929 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
27290ffnd 6504 . . . . . 6 (𝜑𝐴 Fn ℕ)
273 fniunfv 6998 . . . . . 6 (𝐴 Fn ℕ → 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
274272, 273syl 17 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
275270, 271, 2743eqtr2rd 2866 . . . 4 (𝜑 ran 𝐴 = ℝ)
276 eqid 2824 . . . 4 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))
27790, 144, 275, 10, 276itg1climres 24321 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ⇝ (∫1𝐻))
278 nnex 11640 . . . . 5 ℕ ∈ V
279278mptex 6977 . . . 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 6661 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
282281eleq2d 2901 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑥 ∈ (𝐴𝑗) ↔ 𝑥 ∈ (𝐴𝑘)))
283282ifbid 4472 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0) = if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
284283mpteq2dv 5148 . . . . . . . 8 (𝑗 = 𝑘 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
285284fveq2d 6665 . . . . . . 7 (𝑗 = 𝑘 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
286 eqid 2824 . . . . . . 7 (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))
287 fvex 6674 . . . . . . 7 (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ V
288285, 286, 287fvmpt 6759 . . . . . 6 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
289288adantl 485 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
29090ffvelrnda 6842 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐴𝑘) ∈ dom vol)
291 eqid 2824 . . . . . . . 8 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
292291i1fres 24312 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑘) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
29310, 290, 292syl2an2r 684 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
294 itg1cl 24292 . . . . . 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 2916 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℝ)
297296recnd 10667 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℂ)
298285oveq2d 7165 . . . . . 6 (𝑗 = 𝑘 → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
299 eqid 2824 . . . . . 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 7182 . . . . . 6 (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) ∈ V
301298, 299, 300fvmpt 6759 . . . . 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 7165 . . . . 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 2862 . . . 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 485 . . 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 14993 . 2 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ⇝ (𝑇 · (∫1𝐻)))
306 icossicc 12823 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
307 fss 6517 . . . . . . 7 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
3086, 306, 307sylancl 589 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
309 itg2mono.10 . . . . . . 7 (𝜑𝑆 ∈ ℝ)
310309adantr 484 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
311 itg2cl 24339 . . . . . . . . . . 11 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
312308, 311syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
313312fmpttd 6870 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
314313frnd 6510 . . . . . . . 8 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
315 fvex 6674 . . . . . . . . . . 11 (∫2‘(𝐹𝑛)) ∈ V
316315elabrex 6994 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
317316adantl 485 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
318 eqid 2824 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
319318rnmpt 5814 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))}
320317, 319eleqtrrdi 2927 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))))
321 supxrub 12714 . . . . . . . 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 684 . . . . . . 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 5094 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ 𝑆)
325 itg2lecl 24345 . . . . . 6 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ ∧ (∫2‘(𝐹𝑛)) ≤ 𝑆) → (∫2‘(𝐹𝑛)) ∈ ℝ)
326308, 310, 324, 325syl3anc 1368 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ)
327326fmpttd 6870 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ)
328308ralrimiva 3177 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞))
329 fveq2 6661 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
330329feq1d 6488 . . . . . . . . . . 11 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹𝑘):ℝ⟶(0[,]+∞)))
331330cbvralvw 3434 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
332328, 331sylib 221 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
333 peano2nn 11646 . . . . . . . . 9 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
334 fveq2 6661 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
335334feq1d 6488 . . . . . . . . . 10 (𝑘 = (𝑛 + 1) → ((𝐹𝑘):ℝ⟶(0[,]+∞) ↔ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞)))
336335rspccva 3608 . . . . . . . . 9 ((∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞) ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
337332, 333, 336syl2an 598 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
338 itg2le 24346 . . . . . . . 8 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞) ∧ (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
339308, 337, 91, 338syl3anc 1368 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
340339ralrimiva 3177 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
341 2fveq3 6666 . . . . . . . . . 10 (𝑛 = 𝑘 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑘)))
342 fvex 6674 . . . . . . . . . 10 (∫2‘(𝐹𝑘)) ∈ V
343341, 318, 342fvmpt 6759 . . . . . . . . 9 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
344 peano2nn 11646 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
345 2fveq3 6666 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹‘(𝑘 + 1))))
346 fvex 6674 . . . . . . . . . . 11 (∫2‘(𝐹‘(𝑘 + 1))) ∈ V
347345, 318, 346fvmpt 6759 . . . . . . . . . 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 5065 . . . . . . . 8 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
350349ralbiia 3159 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
351 fvoveq1 7172 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
352351fveq2d 6665 . . . . . . . . 9 (𝑛 = 𝑘 → (∫2‘(𝐹‘(𝑛 + 1))) = (∫2‘(𝐹‘(𝑘 + 1))))
353341, 352breq12d 5065 . . . . . . . 8 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
354353cbvralvw 3434 . . . . . . 7 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
355350, 354bitr4i 281 . . . . . 6 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
356340, 355sylibr 237 . . . . 5 (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
357356r19.21bi 3203 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
358324ralrimiva 3177 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆)
359343breq1d 5062 . . . . . . . . 9 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
360359ralbiia 3159 . . . . . . . 8 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
361341breq1d 5062 . . . . . . . . 9 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
362361cbvralvw 3434 . . . . . . . 8 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
363360, 362bitr4i 281 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥)
364 breq2 5056 . . . . . . . 8 (𝑥 = 𝑆 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑛)) ≤ 𝑆))
365364ralbidv 3192 . . . . . . 7 (𝑥 = 𝑆 → (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
366363, 365syl5bb 286 . . . . . 6 (𝑥 = 𝑆 → (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
367366rspcev 3609 . . . . 5 ((𝑆 ∈ ℝ ∧ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
368309, 358, 367syl2anc 587 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
3691, 2, 327, 357, 368climsup 15026 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
370327frnd 6510 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ)
371318, 312dmmptd 6482 . . . . . . 7 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ℕ)
372240a1i 11 . . . . . . 7 (𝜑 → ℕ ≠ ∅)
373371, 372eqnetrd 3081 . . . . . 6 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
374 dm0rn0 5782 . . . . . . 7 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅)
375374necon3bii 3066 . . . . . 6 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
376373, 375sylib 221 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
377315, 318fnmpti 6480 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ
378 breq1 5055 . . . . . . . . 9 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) → (𝑧𝑥 ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
379378ralrn 6845 . . . . . . . 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 3289 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
382368, 381mpbird 260 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥)
383 supxrre 12717 . . . . 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 1368 . . . 4 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
385323, 384syl5req 2872 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ) = 𝑆)
386369, 385breqtrd 5078 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ 𝑆)
38717adantr 484 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
38890ffvelrnda 6842 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ∈ dom vol)
389276i1fres 24312 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑗) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
39010, 388, 389syl2an2r 684 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
391 itg1cl 24292 . . . . . 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 10669 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ∈ ℝ)
394393fmpttd 6870 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))):ℕ⟶ℝ)
395394ffvelrnda 6842 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) ∈ ℝ)
396327ffvelrnda 6842 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ∈ ℝ)
397329feq1d 6488 . . . . . . . 8 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑘):ℝ⟶(0[,)+∞)))
398397cbvralvw 3434 . . . . . . 7 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
39999, 398sylib 221 . . . . . 6 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
400399r19.21bi 3203 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,)+∞))
401 fss 6517 . . . . 5 (((𝐹𝑘):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑘):ℝ⟶(0[,]+∞))
402400, 306, 401sylancl 589 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,]+∞))
40323a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ℝ ∈ V)
40417adantr 484 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℝ)
405404adantr 484 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
406 fvex 6674 . . . . . . . . 9 (𝐻𝑥) ∈ V
407 c0ex 10633 . . . . . . . . 9 0 ∈ V
408406, 407ifex 4498 . . . . . . . 8 if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V
409408a1i 11 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V)
410 fconstmpt 5601 . . . . . . . 8 (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇)
411410a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇))
412 eqidd 2825 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
413403, 405, 409, 411, 412offval2 7420 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
414 ovif2 7245 . . . . . . . 8 (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0))
41547adantr 484 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℂ)
416415mul01d 10837 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝑇 · 0) = 0)
417416ifeq2d 4469 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
418414, 417syl5eq 2871 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
419418mpteq2dv 5148 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
420413, 419eqtrd 2859 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
421293, 404i1fmulc 24310 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ dom ∫1)
422420, 421eqeltrrd 2917 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∈ dom ∫1)
423 iftrue 4456 . . . . . . . . 9 (𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
424423adantl 485 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
425329fveq1d 6663 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑘)‘𝑥))
426425breq2d 5064 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
427426rabbidv 3465 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
42823rabex 5221 . . . . . . . . . . . . 13 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ∈ V
429427, 89, 428fvmpt 6759 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
430429ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
431430eleq2d 2901 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐴𝑘) ↔ 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)}))
432431biimpa 480 . . . . . . . . 9 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
433 rabid 3369 . . . . . . . . . 10 (𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ↔ (𝑥 ∈ ℝ ∧ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
434433simprbi 500 . . . . . . . . 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 5074 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
437 iffalse 4459 . . . . . . . . 9 𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
438437adantl 485 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
439400ffvelrnda 6842 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ (0[,)+∞))
440 elrege0 12841 . . . . . . . . . . 11 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑘)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑘)‘𝑥)))
441440simprbi 500 . . . . . . . . . 10 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) → 0 ≤ ((𝐹𝑘)‘𝑥))
442439, 441syl 17 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹𝑘)‘𝑥))
443442adantr 484 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → 0 ≤ ((𝐹𝑘)‘𝑥))
444438, 443eqbrtrd 5074 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
445436, 444pm2.61dan 812 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
446445ralrimiva 3177 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
447 ovex 7182 . . . . . . . 8 (𝑇 · (𝐻𝑥)) ∈ V
448447, 407ifex 4498 . . . . . . 7 if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V
449448a1i 11 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V)
450 fvexd 6676 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ V)
451 eqidd 2825 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
452400feqmptd 6724 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = (𝑥 ∈ ℝ ↦ ((𝐹𝑘)‘𝑥)))
453403, 449, 450, 451, 452ofrfval2 7421 . . . . 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 260 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘r ≤ (𝐹𝑘))
455 itg2ub 24340 . . . 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 1368 . . 3 ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))) ≤ (∫2‘(𝐹𝑘)))
457301adantl 485 . . . 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 24311 . . . 4 ((𝜑𝑘 ∈ ℕ) → (∫1‘((ℝ × {𝑇}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
459420fveq2d 6665 . . . 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 2865 . . 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 485 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
462456, 460, 4613brtr4d 5084 . 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 14996 1 (𝜑 → (𝑇 · (∫1𝐻)) ≤ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  {cab 2802  wne 3014  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  cdif 3916  cin 3918  wss 3919  c0 4276  ifcif 4450  {csn 4550   cuni 4824   ciun 4905   class class class wbr 5052  cmpt 5132   × cxp 5540  ccnv 5541  dom cdm 5542  ran crn 5543  cima 5545   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  f cof 7401  r cofr 7402  supcsup 8901  cc 10533  cr 10534  0cc0 10535  1c1 10536   + caddc 10538   · cmul 10540  +∞cpnf 10670  -∞cmnf 10671  *cxr 10672   < clt 10673  cle 10674  cmin 10868  -cneg 10869  cn 11634  (,)cioo 12735  [,)cico 12737  [,]cicc 12738  cli 14841  volcvol 24070  MblFncmbf 24221  1citg1 24222  2citg2 24223
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cc 9855  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-addf 10614
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-disj 5018  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-omul 8103  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fi 8872  df-sup 8903  df-inf 8904  df-oi 8971  df-dju 9327  df-card 9365  df-acn 9368  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ioo 12739  df-ioc 12740  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-rest 16696  df-topgen 16717  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-ovol 24071  df-vol 24072  df-mbf 24226  df-itg1 24227  df-itg2 24228
This theorem is referenced by:  itg2monolem3  24359
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