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Theorem itg2monolem1 23562
 Description: Lemma for itg2mono 23565. We show that for any constant 𝑡 less than one, 𝑡 · ∫1𝐻 is less than 𝑆, and so ∫1𝐻 ≤ 𝑆, which is one half of the equality in itg2mono 23565. Consider the sequence 𝐴(𝑛) = {𝑥 ∣ 𝑡 · 𝐻 ≤ 𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is ℝ, and so 𝐻 ↾ 𝐴(𝑛) has an integral which equals ∫1𝐻 in the limit, by itg1climres 23526. Then by taking the limit in (𝑡 · 𝐻) ↾ 𝐴(𝑛) ≤ 𝐹(𝑛), we get 𝑡 · ∫1𝐻 ≤ 𝑆 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 11761 . 2 ℕ = (ℤ‘1)
2 1zzd 11446 . 2 (𝜑 → 1 ∈ ℤ)
3 readdcl 10057 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
43adantl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
5 itg2mono.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
6 rge0ssre 12318 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ
7 fss 6094 . . . . . . . . . . . . . . . 16 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
85, 6, 7sylancl 695 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
9 itg2mono.8 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻 ∈ dom ∫1)
10 itg2mono.7 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑇 ∈ (0(,)1))
11 0xr 10124 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ*
12 1re 10077 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
1312rexri 10135 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ*
14 elioo2 12254 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1)))
1511, 13, 14mp2an 708 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1610, 15sylib 208 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1716simp1d 1093 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ℝ)
1817renegcld 10495 . . . . . . . . . . . . . . . . . 18 (𝜑 → -𝑇 ∈ ℝ)
199, 18i1fmulc 23515 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ dom ∫1)
2019adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ dom ∫1)
21 i1ff 23488 . . . . . . . . . . . . . . . 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 10065 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2423a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ℝ ∈ V)
25 inidm 3855 . . . . . . . . . . . . . . 15 (ℝ ∩ ℝ) = ℝ
264, 8, 22, 24, 24, 25off 6954 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)):ℝ⟶ℝ)
2726adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)):ℝ⟶ℝ)
28 ffn 6083 . . . . . . . . . . . . 13 (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)):ℝ⟶ℝ → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) Fn ℝ)
2927, 28syl 17 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) Fn ℝ)
30 elpreima 6377 . . . . . . . . . . . 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))))
3129, 30syl 17 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ↔ (𝑥 ∈ ℝ ∧ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0))))
32 simpr 476 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
3332biantrurd 528 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ (𝑥 ∈ ℝ ∧ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0))))
3431, 33bitr4d 271 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ↔ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0)))
3526ffvelrnda 6399 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ ℝ)
3635biantrurd 528 . . . . . . . . . . 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)))
37 elioomnf 12306 . . . . . . . . . . . 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)))
3811, 37ax-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))
3936, 38syl6rbbr 279 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) < 0))
40 ffn 6083 . . . . . . . . . . . . . . 15 ((𝐹𝑛):ℝ⟶ℝ → (𝐹𝑛) Fn ℝ)
418, 40syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) Fn ℝ)
42 ffn 6083 . . . . . . . . . . . . . . 15 (((ℝ × {-𝑇}) ∘𝑓 · 𝐻):ℝ⟶ℝ → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) Fn ℝ)
4322, 42syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) Fn ℝ)
44 eqidd 2652 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
4518adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → -𝑇 ∈ ℝ)
46 i1ff 23488 . . . . . . . . . . . . . . . . . . 19 (𝐻 ∈ dom ∫1𝐻:ℝ⟶ℝ)
479, 46syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻:ℝ⟶ℝ)
48 ffn 6083 . . . . . . . . . . . . . . . . . 18 (𝐻:ℝ⟶ℝ → 𝐻 Fn ℝ)
4947, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐻 Fn ℝ)
5049adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → 𝐻 Fn ℝ)
51 eqidd 2652 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) = (𝐻𝑥))
5224, 45, 50, 51ofc1 6962 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘𝑓 · 𝐻)‘𝑥) = (-𝑇 · (𝐻𝑥)))
5317recnd 10106 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 ∈ ℂ)
5453ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ)
5547ffvelrnda 6399 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5655adantlr 751 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5756recnd 10106 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℂ)
5854, 57mulneg1d 10521 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (-𝑇 · (𝐻𝑥)) = -(𝑇 · (𝐻𝑥)))
5952, 58eqtrd 2685 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘𝑓 · 𝐻)‘𝑥) = -(𝑇 · (𝐻𝑥)))
6041, 43, 24, 24, 25, 44, 59ofval 6948 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))))
618ffvelrnda 6399 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
6261recnd 10106 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℂ)
6317adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
6463, 55remulcld 10108 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
6564adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
6665recnd 10106 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℂ)
6762, 66negsubd 10436 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6860, 67eqtrd 2685 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6968breq1d 4695 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) < 0 ↔ (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0))
70 0red 10079 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
7161, 65, 70ltsubaddd 10661 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0 ↔ ((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥)))))
7266addid2d 10275 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 + (𝑇 · (𝐻𝑥))) = (𝑇 · (𝐻𝑥)))
7372breq2d 4697 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥))) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7469, 71, 733bitrd 294 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) < 0 ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7534, 39, 743bitrd 294 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7675notbid 307 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
77 eldif 3617 . . . . . . . . . 10 (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))))
7877baib 964 . . . . . . . . 9 (𝑥 ∈ ℝ → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))))
7978adantl 481 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))))
8065, 61lenltd 10221 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
8176, 79, 803bitr4d 300 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)))
8281rabbi2dva 3854 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)))) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
83 rembl 23354 . . . . . . 7 ℝ ∈ dom vol
84 itg2mono.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
85 i1fmbf 23487 . . . . . . . . . . 11 (((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ dom ∫1 → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ MblFn)
8620, 85syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ MblFn)
8784, 86mbfadd 23473 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) ∈ MblFn)
88 mbfima 23444 . . . . . . . . 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)
8987, 26, 88syl2anc 694 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ∈ dom vol)
90 cmmbl 23348 . . . . . . . 8 ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ∈ dom vol → (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ∈ dom vol)
9189, 90syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ∈ dom vol)
92 inmbl 23356 . . . . . . 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)
9383, 91, 92sylancr 696 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)))) ∈ dom vol)
9482, 93eqeltrrd 2731 . . . . 5 ((𝜑𝑛 ∈ ℕ) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} ∈ dom vol)
95 itg2mono.11 . . . . 5 𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
9694, 95fmptd 6425 . . . 4 (𝜑𝐴:ℕ⟶dom vol)
97 itg2mono.4 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
9897ralrimiva 2995 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
99 fveq2 6229 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
100 oveq1 6697 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
101100fveq2d 6233 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑗 + 1)))
10299, 101breq12d 4698 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)) ↔ (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1))))
103102cbvralv 3201 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)))
10498, 103sylib 208 . . . . . . . . . 10 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)))
105104r19.21bi 2961 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)))
1065ralrimiva 2995 . . . . . . . . . . . . 13 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
10799feq1d 6068 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑗):ℝ⟶(0[,)+∞)))
108107cbvralv 3201 . . . . . . . . . . . . 13 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
109106, 108sylib 208 . . . . . . . . . . . 12 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
110109r19.21bi 2961 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
111 ffn 6083 . . . . . . . . . . 11 ((𝐹𝑗):ℝ⟶(0[,)+∞) → (𝐹𝑗) Fn ℝ)
112110, 111syl 17 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) Fn ℝ)
113 peano2nn 11070 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
114 fveq2 6229 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (𝐹𝑛) = (𝐹‘(𝑗 + 1)))
115114feq1d 6068 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞)))
116115rspccva 3339 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (𝑗 + 1) ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
117106, 113, 116syl2an 493 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
118 ffn 6083 . . . . . . . . . . 11 ((𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞) → (𝐹‘(𝑗 + 1)) Fn ℝ)
119117, 118syl 17 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) Fn ℝ)
12023a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ℝ ∈ V)
121 eqidd 2652 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑥))
122 eqidd 2652 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
123112, 119, 120, 120, 25, 121, 122ofrfval 6947 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)) ↔ ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
124105, 123mpbid 222 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
125124r19.21bi 2961 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
12617ad2antrr 762 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
12747adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 𝐻:ℝ⟶ℝ)
128127ffvelrnda 6399 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
129126, 128remulcld 10108 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
130 fss 6094 . . . . . . . . . 10 (((𝐹𝑗):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑗):ℝ⟶ℝ)
131110, 6, 130sylancl 695 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶ℝ)
132131ffvelrnda 6399 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
133 fss 6094 . . . . . . . . . 10 (((𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
134117, 6, 133sylancl 695 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
135134ffvelrnda 6399 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) ∈ ℝ)
136 letr 10169 . . . . . . . 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)))
137129, 132, 135, 136syl3anc 1366 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) ∧ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
138125, 137mpan2d 710 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
139138ss2rabdv 3716 . . . . 5 ((𝜑𝑗 ∈ ℕ) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ⊆ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
14099fveq1d 6231 . . . . . . . . 9 (𝑛 = 𝑗 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑗)‘𝑥))
141140breq2d 4697 . . . . . . . 8 (𝑛 = 𝑗 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
142141rabbidv 3220 . . . . . . 7 (𝑛 = 𝑗 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
14323rabex 4845 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ∈ V
144142, 95, 143fvmpt 6321 . . . . . 6 (𝑗 ∈ ℕ → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
145144adantl 481 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
146113adantl 481 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
147114fveq1d 6231 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐹𝑛)‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
148147breq2d 4697 . . . . . . . 8 (𝑛 = (𝑗 + 1) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
149148rabbidv 3220 . . . . . . 7 (𝑛 = (𝑗 + 1) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
15023rabex 4845 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)} ∈ V
151149, 95, 150fvmpt 6321 . . . . . 6 ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
152146, 151syl 17 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐴‘(𝑗 + 1)) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
153139, 145, 1523sstr4d 3681 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ⊆ (𝐴‘(𝑗 + 1)))
15464adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
15555adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℝ)
15661an32s 863 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
157 eqid 2651 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
158156, 157fmptd 6425 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
159 frn 6091 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
160158, 159syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
161 1nn 11069 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
162157, 156dmmptd 6062 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
163161, 162syl5eleqr 2737 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
164 ne0i 3954 . . . . . . . . . . . . . . . . . 18 (1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
165163, 164syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
166 dm0rn0 5374 . . . . . . . . . . . . . . . . . 18 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
167166necon3bii 2875 . . . . . . . . . . . . . . . . 17 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
168165, 167sylib 208 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
169 itg2mono.5 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
170 ffn 6083 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
171158, 170syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
172 breq1 4688 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
173172ralrn 6402 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
174171, 173syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
175 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
176175fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
177 fvex 6239 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑚)‘𝑥) ∈ V
178176, 157, 177fvmpt 6321 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
179178breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
180179ralbiia 3008 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
181176breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
182181cbvralv 3201 . . . . . . . . . . . . . . . . . . . 20 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
183180, 182bitr4i 267 . . . . . . . . . . . . . . . . . . 19 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
184174, 183syl6bb 276 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
185184rexbidv 3081 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
186169, 185mpbird 247 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
187 suprcl 11021 . . . . . . . . . . . . . . . 16 ((ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
188160, 168, 186, 187syl3anc 1366 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
189188adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
19016simp3d 1095 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 < 1)
191190adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 < 1)
19217adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 ∈ ℝ)
193 1red 10093 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 1 ∈ ℝ)
194 simprr 811 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 0 < (𝐻𝑥))
195 ltmul1 10911 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝐻𝑥) ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
196192, 193, 155, 194, 195syl112anc 1370 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
197191, 196mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥)))
198155recnd 10106 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℂ)
199198mulid2d 10096 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (1 · (𝐻𝑥)) = (𝐻𝑥))
200197, 199breqtrd 4711 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < (𝐻𝑥))
201 itg2mono.9 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻𝑟𝐺)
202 itg2mono.1 . . . . . . . . . . . . . . . . . . . . 21 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
203188, 202fmptd 6425 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:ℝ⟶ℝ)
204 ffn 6083 . . . . . . . . . . . . . . . . . . . 20 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
205203, 204syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 Fn ℝ)
20623a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ℝ ∈ V)
207 eqidd 2652 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐻𝑦) = (𝐻𝑦))
208 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑦 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑦))
209208mpteq2dv 4778 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
210209rneqd 5385 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
211210supeq1d 8393 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
212 ltso 10156 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
213212supex 8410 . . . . . . . . . . . . . . . . . . . . 21 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ) ∈ V
214211, 202, 213fvmpt 6321 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℝ → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
215214adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
21649, 205, 206, 206, 25, 207, 215ofrfval 6947 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐻𝑟𝐺 ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
217201, 216mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
218 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐻𝑥) = (𝐻𝑦))
219218, 211breq12d 4698 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
220219cbvralv 3201 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
221217, 220sylibr 224 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
222221r19.21bi 2961 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
223222adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
224154, 155, 189, 200, 223ltletrd 10235 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
225160adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
226168adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
227186adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
228 suprlub 11025 . . . . . . . . . . . . . 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))
229225, 226, 227, 154, 228syl31anc 1369 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ((𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤))
230224, 229mpbid 222 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤)
231171adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
232 breq2 4689 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) → ((𝑇 · (𝐻𝑥)) < 𝑤 ↔ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗)))
233232rexrn 6401 . . . . . . . . . . . . . 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)))
234231, 233syl 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)))
235 fvex 6239 . . . . . . . . . . . . . . . 16 ((𝐹𝑗)‘𝑥) ∈ V
236140, 157, 235fvmpt 6321 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) = ((𝐹𝑗)‘𝑥))
237236breq2d 4697 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ((𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
238237rexbiia 3069 . . . . . . . . . . . . 13 (∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
239234, 238syl6bb 276 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤 ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
240230, 239mpbid 222 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
241192, 155remulcld 10108 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
242241adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
243110adantlr 751 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
244 simplr 807 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑥 ∈ ℝ)
245243, 244ffvelrnd 6400 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ (0[,)+∞))
246 elrege0 12316 . . . . . . . . . . . . . . . 16 (((𝐹𝑗)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
247245, 246sylib 208 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
248247simpld 474 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
249248adantlrr 757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
250 ltle 10164 . . . . . . . . . . . . 13 (((𝑇 · (𝐻𝑥)) ∈ ℝ ∧ ((𝐹𝑗)‘𝑥) ∈ ℝ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
251242, 249, 250syl2anc 694 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
252251reximdva 3046 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
253240, 252mpd 15 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
254253anassrs 681 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 0 < (𝐻𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
255161ne0ii 3956 . . . . . . . . . . 11 ℕ ≠ ∅
25664adantrr 753 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
257256adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
258 0red 10079 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ)
259247adantlrr 757 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
260259simpld 474 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
261 simplrr 818 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ≤ 0)
26255adantrr 753 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝐻𝑥) ∈ ℝ)
263262adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ∈ ℝ)
26417ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
26516simp2d 1094 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑇)
266265ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 < 𝑇)
267 lemul2 10914 . . . . . . . . . . . . . . . 16 (((𝐻𝑥) ∈ ℝ ∧ 0 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
268263, 258, 264, 266, 267syl112anc 1370 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
269261, 268mpbid 222 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0))
270264recnd 10106 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℂ)
271270mul01d 10273 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · 0) = 0)
272269, 271breqtrd 4711 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ 0)
273259simprd 478 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ≤ ((𝐹𝑗)‘𝑥))
274257, 258, 260, 272, 273letrd 10232 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
275274ralrimiva 2995 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
276 r19.2z 4093 . . . . . . . . . . 11 ((ℕ ≠ ∅ ∧ ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
277255, 275, 276sylancr 696 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
278277anassrs 681 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝐻𝑥) ≤ 0) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
279 0red 10079 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
280254, 278, 279, 55ltlecasei 10183 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
281280ralrimiva 2995 . . . . . . 7 (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
282 rabid2 3148 . . . . . . 7 (ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
283281, 282sylibr 224 . . . . . 6 (𝜑 → ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
284 iunrab 4599 . . . . . 6 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)}
285283, 284syl6eqr 2703 . . . . 5 (𝜑 → ℝ = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
286145iuneq2dv 4574 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
287 ffn 6083 . . . . . . 7 (𝐴:ℕ⟶dom vol → 𝐴 Fn ℕ)
28896, 287syl 17 . . . . . 6 (𝜑𝐴 Fn ℕ)
289 fniunfv 6545 . . . . . 6 (𝐴 Fn ℕ → 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
290288, 289syl 17 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
291285, 286, 2903eqtr2rd 2692 . . . 4 (𝜑 ran 𝐴 = ℝ)
292 eqid 2651 . . . 4 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))
29396, 153, 291, 9, 292itg1climres 23526 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ⇝ (∫1𝐻))
294 nnex 11064 . . . . 5 ℕ ∈ V
295294mptex 6527 . . . 4 (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ∈ V
296295a1i 11 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ∈ V)
297 fveq2 6229 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
298297eleq2d 2716 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑥 ∈ (𝐴𝑗) ↔ 𝑥 ∈ (𝐴𝑘)))
299298ifbid 4141 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0) = if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
300299mpteq2dv 4778 . . . . . . . 8 (𝑗 = 𝑘 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
301300fveq2d 6233 . . . . . . 7 (𝑗 = 𝑘 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
302 eqid 2651 . . . . . . 7 (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))
303 fvex 6239 . . . . . . 7 (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ V
304301, 302, 303fvmpt 6321 . . . . . 6 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
305304adantl 481 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
3069adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → 𝐻 ∈ dom ∫1)
30796ffvelrnda 6399 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐴𝑘) ∈ dom vol)
308 eqid 2651 . . . . . . . 8 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
309308i1fres 23517 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑘) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
310306, 307, 309syl2anc 694 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
311 itg1cl 23497 . . . . . 6 ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ ℝ)
312310, 311syl 17 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ ℝ)
313305, 312eqeltrd 2730 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℝ)
314313recnd 10106 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℂ)
315301oveq2d 6706 . . . . . 6 (𝑗 = 𝑘 → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
316 eqid 2651 . . . . . 6 (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) = (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))
317 ovex 6718 . . . . . 6 (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) ∈ V
318315, 316, 317fvmpt 6321 . . . . 5 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
319304oveq2d 6706 . . . . 5 (𝑘 ∈ ℕ → (𝑇 · ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘)) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
320318, 319eqtr4d 2688 . . . 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)))
321320adantl 481 . . 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)))
3221, 2, 293, 53, 296, 314, 321climmulc2 14411 . 2 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ⇝ (𝑇 · (∫1𝐻)))
323 icossicc 12298 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
324 fss 6094 . . . . . . 7 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
3255, 323, 324sylancl 695 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
326 itg2mono.10 . . . . . . 7 (𝜑𝑆 ∈ ℝ)
327326adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
328 itg2cl 23544 . . . . . . . . . . . 12 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
329325, 328syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
330 eqid 2651 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
331329, 330fmptd 6425 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
332 frn 6091 . . . . . . . . . 10 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ* → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
333331, 332syl 17 . . . . . . . . 9 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
334333adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
335 fvex 6239 . . . . . . . . . . 11 (∫2‘(𝐹𝑛)) ∈ V
336335elabrex 6541 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
337336adantl 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
338330rnmpt 5403 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))}
339337, 338syl6eleqr 2741 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))))
340 supxrub 12192 . . . . . . . 8 ((ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ* ∧ (∫2‘(𝐹𝑛)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))) → (∫2‘(𝐹𝑛)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ))
341334, 339, 340syl2anc 694 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ))
342 itg2mono.6 . . . . . . 7 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )
343341, 342syl6breqr 4727 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ 𝑆)
344 itg2lecl 23550 . . . . . 6 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ ∧ (∫2‘(𝐹𝑛)) ≤ 𝑆) → (∫2‘(𝐹𝑛)) ∈ ℝ)
345325, 327, 343, 344syl3anc 1366 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ)
346345, 330fmptd 6425 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ)
347325ralrimiva 2995 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞))
348 fveq2 6229 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
349348feq1d 6068 . . . . . . . . . . 11 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹𝑘):ℝ⟶(0[,]+∞)))
350349cbvralv 3201 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
351347, 350sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
352 peano2nn 11070 . . . . . . . . 9 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
353 fveq2 6229 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
354353feq1d 6068 . . . . . . . . . 10 (𝑘 = (𝑛 + 1) → ((𝐹𝑘):ℝ⟶(0[,]+∞) ↔ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞)))
355354rspccva 3339 . . . . . . . . 9 ((∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞) ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
356351, 352, 355syl2an 493 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
357 itg2le 23551 . . . . . . . 8 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞) ∧ (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1))) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
358325, 356, 97, 357syl3anc 1366 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
359358ralrimiva 2995 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
360348fveq2d 6233 . . . . . . . . . 10 (𝑛 = 𝑘 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑘)))
361 fvex 6239 . . . . . . . . . 10 (∫2‘(𝐹𝑘)) ∈ V
362360, 330, 361fvmpt 6321 . . . . . . . . 9 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
363 peano2nn 11070 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
364 fveq2 6229 . . . . . . . . . . . 12 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
365364fveq2d 6233 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹‘(𝑘 + 1))))
366 fvex 6239 . . . . . . . . . . 11 (∫2‘(𝐹‘(𝑘 + 1))) ∈ V
367365, 330, 366fvmpt 6321 . . . . . . . . . 10 ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) = (∫2‘(𝐹‘(𝑘 + 1))))
368363, 367syl 17 . . . . . . . . 9 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) = (∫2‘(𝐹‘(𝑘 + 1))))
369362, 368breq12d 4698 . . . . . . . 8 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
370369ralbiia 3008 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
371 oveq1 6697 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
372371fveq2d 6233 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
373372fveq2d 6233 . . . . . . . . 9 (𝑛 = 𝑘 → (∫2‘(𝐹‘(𝑛 + 1))) = (∫2‘(𝐹‘(𝑘 + 1))))
374360, 373breq12d 4698 . . . . . . . 8 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
375374cbvralv 3201 . . . . . . 7 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
376370, 375bitr4i 267 . . . . . 6 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
377359, 376sylibr 224 . . . . 5 (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
378377r19.21bi 2961 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
379343ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆)
380362breq1d 4695 . . . . . . . . 9 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
381380ralbiia 3008 . . . . . . . 8 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
382360breq1d 4695 . . . . . . . . 9 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
383382cbvralv 3201 . . . . . . . 8 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
384381, 383bitr4i 267 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥)
385 breq2 4689 . . . . . . . 8 (𝑥 = 𝑆 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑛)) ≤ 𝑆))
386385ralbidv 3015 . . . . . . 7 (𝑥 = 𝑆 → (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
387384, 386syl5bb 272 . . . . . 6 (𝑥 = 𝑆 → (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
388387rspcev 3340 . . . . 5 ((𝑆 ∈ ℝ ∧ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
389326, 379, 388syl2anc 694 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
3901, 2, 346, 378, 389climsup 14444 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
391 frn 6091 . . . . . 6 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ)
392346, 391syl 17 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ)
393330, 329dmmptd 6062 . . . . . . 7 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ℕ)
394255a1i 11 . . . . . . 7 (𝜑 → ℕ ≠ ∅)
395393, 394eqnetrd 2890 . . . . . 6 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
396 dm0rn0 5374 . . . . . . 7 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅)
397396necon3bii 2875 . . . . . 6 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
398395, 397sylib 208 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
399335, 330fnmpti 6060 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ
400399a1i 11 . . . . . . . 8 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ)
401 breq1 4688 . . . . . . . . 9 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) → (𝑧𝑥 ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
402401ralrn 6402 . . . . . . . 8 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
403400, 402syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
404403rexbidv 3081 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
405389, 404mpbird 247 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥)
406 supxrre 12195 . . . . 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, < ))
407392, 398, 405, 406syl3anc 1366 . . . 4 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
408342, 407syl5req 2698 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ) = 𝑆)
409390, 408breqtrd 4711 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ 𝑆)
41017adantr 480 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
4119adantr 480 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝐻 ∈ dom ∫1)
41296ffvelrnda 6399 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ∈ dom vol)
413292i1fres 23517 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑗) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
414411, 412, 413syl2anc 694 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
415 itg1cl 23497 . . . . . 6 ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) ∈ ℝ)
416414, 415syl 17 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) ∈ ℝ)
417410, 416remulcld 10108 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ∈ ℝ)
418417, 316fmptd 6425 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))):ℕ⟶ℝ)
419418ffvelrnda 6399 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) ∈ ℝ)
420346ffvelrnda 6399 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ∈ ℝ)
421348feq1d 6068 . . . . . . . 8 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑘):ℝ⟶(0[,)+∞)))
422421cbvralv 3201 . . . . . . 7 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
423106, 422sylib 208 . . . . . 6 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
424423r19.21bi 2961 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,)+∞))
425 fss 6094 . . . . 5 (((𝐹𝑘):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑘):ℝ⟶(0[,]+∞))
426424, 323, 425sylancl 695 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,]+∞))
42723a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ℝ ∈ V)
42817adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℝ)
429428adantr 480 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
430 fvex 6239 . . . . . . . . 9 (𝐻𝑥) ∈ V
431 c0ex 10072 . . . . . . . . 9 0 ∈ V
432430, 431ifex 4189 . . . . . . . 8 if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V
433432a1i 11 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V)
434 fconstmpt 5197 . . . . . . . 8 (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇)
435434a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇))
436 eqidd 2652 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
437427, 429, 433, 435, 436offval2 6956 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
438 ovif2 6780 . . . . . . . 8 (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0))
43953adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℂ)
440439mul01d 10273 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝑇 · 0) = 0)
441440ifeq2d 4138 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
442438, 441syl5eq 2697 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
443442mpteq2dv 4778 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
444437, 443eqtrd 2685 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
445310, 428i1fmulc 23515 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ dom ∫1)
446444, 445eqeltrrd 2731 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∈ dom ∫1)
447 iftrue 4125 . . . . . . . . 9 (𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
448447adantl 481 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
449348fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑘)‘𝑥))
450449breq2d 4697 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
451450rabbidv 3220 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
45223rabex 4845 . . . . . . . . . . . . 13 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ∈ V
453451, 95, 452fvmpt 6321 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
454453ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
455454eleq2d 2716 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐴𝑘) ↔ 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)}))
456455biimpa 500 . . . . . . . . 9 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
457 rabid 3145 . . . . . . . . . 10 (𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ↔ (𝑥 ∈ ℝ ∧ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
458457simprbi 479 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥))
459456, 458syl 17 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥))
460448, 459eqbrtrd 4707 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
461 iffalse 4128 . . . . . . . . 9 𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
462461adantl 481 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
463424ffvelrnda 6399 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ (0[,)+∞))
464 elrege0 12316 . . . . . . . . . . 11 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑘)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑘)‘𝑥)))
465464simprbi 479 . . . . . . . . . 10 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) → 0 ≤ ((𝐹𝑘)‘𝑥))
466463, 465syl 17 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹𝑘)‘𝑥))
467466adantr 480 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → 0 ≤ ((𝐹𝑘)‘𝑥))
468462, 467eqbrtrd 4707 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
469460, 468pm2.61dan 849 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
470469ralrimiva 2995 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
471 ovex 6718 . . . . . . . 8 (𝑇 · (𝐻𝑥)) ∈ V
472471, 431ifex 4189 . . . . . . 7 if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V
473472a1i 11 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V)
474 fvexd 6241 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ V)
475 eqidd 2652 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
476424feqmptd 6288 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = (𝑥 ∈ ℝ ↦ ((𝐹𝑘)‘𝑥)))
477427, 473, 474, 475, 476ofrfval2 6957 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘𝑟 ≤ (𝐹𝑘) ↔ ∀𝑥 ∈ ℝ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥)))
478470, 477mpbird 247 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘𝑟 ≤ (𝐹𝑘))
479 itg2ub 23545 . . . 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)))
480426, 446, 478, 479syl3anc 1366 . . 3 ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))) ≤ (∫2‘(𝐹𝑘)))
481318adantl 481 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
482310, 428itg1mulc 23516 . . . 4 ((𝜑𝑘 ∈ ℕ) → (∫1‘((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
483444fveq2d 6233 . . . 4 ((𝜑𝑘 ∈ ℕ) → (∫1‘((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))))
484481, 482, 4833eqtr2d 2691 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))))
485362adantl 481 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
486480, 484, 4853brtr4d 4717 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘))
4871, 2, 322, 409, 419, 420, 486climle 14414 1 (𝜑 → (𝑇 · (∫1𝐻)) ≤ 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ifcif 4119  {csn 4210  ∪ cuni 4468  ∪ ciun 4552   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  ◡ccnv 5142  dom cdm 5143  ran crn 5144   “ cima 5146   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937   ∘𝑟 cofr 6938  supcsup 8387  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  -∞cmnf 10110  ℝ*cxr 10111   < clt 10112   ≤ cle 10113   − cmin 10304  -cneg 10305  ℕcn 11058  (,)cioo 12213  [,)cico 12215  [,]cicc 12216   ⇝ cli 14259  volcvol 23278  MblFncmbf 23428  ∫1citg1 23429  ∫2citg2 23430 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435 This theorem is referenced by:  itg2monolem3  23564
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