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Theorem itg2monolem1 23731
Description: Lemma for itg2mono 23734. 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 23734. 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 23695. 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 11941 . 2 ℕ = (ℤ‘1)
2 1zzd 11674 . 2 (𝜑 → 1 ∈ ℤ)
3 readdcl 10304 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
43adantl 469 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
5 itg2mono.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))
6 rge0ssre 12500 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ
7 fss 6269 . . . . . . . . . . . . . . . 16 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑛):ℝ⟶ℝ)
85, 6, 7sylancl 576 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶ℝ)
9 itg2mono.8 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻 ∈ dom ∫1)
10 itg2mono.7 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑇 ∈ (0(,)1))
11 0xr 10371 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ*
12 1re 10325 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
1312rexri 10382 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ*
14 elioo2 12434 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1)))
1511, 13, 14mp2an 675 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 ∈ (0(,)1) ↔ (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1610, 15sylib 209 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇𝑇 < 1))
1716simp1d 1165 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ℝ)
1817renegcld 10742 . . . . . . . . . . . . . . . . . 18 (𝜑 → -𝑇 ∈ ℝ)
199, 18i1fmulc 23684 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ dom ∫1)
2019adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ dom ∫1)
21 i1ff 23657 . . . . . . . . . . . . . . . 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 10312 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2423a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ℝ ∈ V)
25 inidm 4019 . . . . . . . . . . . . . . 15 (ℝ ∩ ℝ) = ℝ
264, 8, 22, 24, 24, 25off 7142 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)):ℝ⟶ℝ)
2726adantr 468 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)):ℝ⟶ℝ)
2827ffnd 6257 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) Fn ℝ)
29 elpreima 6559 . . . . . . . . . . . 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))))
31 simpr 473 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
3231biantrurd 524 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ (𝑥 ∈ ℝ ∧ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0))))
3330, 32bitr4d 273 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ↔ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0)))
3426ffvelrnda 6581 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ ℝ)
3534biantrurd 524 . . . . . . . . . . 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)))
36 elioomnf 12487 . . . . . . . . . . . 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)))
3711, 36ax-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))
3835, 37syl6rbbr 281 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) ∈ (-∞(,)0) ↔ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) < 0))
395ffnd 6257 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) Fn ℝ)
4022ffnd 6257 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) Fn ℝ)
41 eqidd 2807 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
4218adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → -𝑇 ∈ ℝ)
43 i1ff 23657 . . . . . . . . . . . . . . . . . . 19 (𝐻 ∈ dom ∫1𝐻:ℝ⟶ℝ)
449, 43syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻:ℝ⟶ℝ)
4544ffnd 6257 . . . . . . . . . . . . . . . . 17 (𝜑𝐻 Fn ℝ)
4645adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → 𝐻 Fn ℝ)
47 eqidd 2807 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) = (𝐻𝑥))
4824, 42, 46, 47ofc1 7150 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘𝑓 · 𝐻)‘𝑥) = (-𝑇 · (𝐻𝑥)))
4917recnd 10353 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 ∈ ℂ)
5049ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ)
5144ffvelrnda 6581 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5251adantlr 697 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
5352recnd 10353 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℂ)
5450, 53mulneg1d 10768 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (-𝑇 · (𝐻𝑥)) = -(𝑇 · (𝐻𝑥)))
5548, 54eqtrd 2840 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((ℝ × {-𝑇}) ∘𝑓 · 𝐻)‘𝑥) = -(𝑇 · (𝐻𝑥)))
5639, 40, 24, 24, 25, 41, 55ofval 7136 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))))
578ffvelrnda 6581 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
5857recnd 10353 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑛)‘𝑥) ∈ ℂ)
5917adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
6059, 51remulcld 10355 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
6160adantlr 697 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
6261recnd 10353 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℂ)
6358, 62negsubd 10683 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) + -(𝑇 · (𝐻𝑥))) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6456, 63eqtrd 2840 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) = (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))))
6564breq1d 4854 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) < 0 ↔ (((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0))
66 0red 10328 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
6757, 61, 66ltsubaddd 10908 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛)‘𝑥) − (𝑇 · (𝐻𝑥))) < 0 ↔ ((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥)))))
6862addid2d 10522 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 + (𝑇 · (𝐻𝑥))) = (𝑇 · (𝐻𝑥)))
6968breq2d 4856 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑛)‘𝑥) < (0 + (𝑇 · (𝐻𝑥))) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7065, 67, 693bitrd 296 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻))‘𝑥) < 0 ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7133, 38, 703bitrd 296 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ↔ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7271notbid 309 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
73 eldif 3779 . . . . . . . . . 10 (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))))
7473baib 527 . . . . . . . . 9 (𝑥 ∈ ℝ → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))))
7574adantl 469 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ ¬ 𝑥 ∈ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))))
7661, 57lenltd 10468 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ ¬ ((𝐹𝑛)‘𝑥) < (𝑇 · (𝐻𝑥))))
7772, 75, 763bitr4d 302 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)))
7877rabbi2dva 4018 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)))) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
79 rembl 23521 . . . . . . 7 ℝ ∈ dom vol
80 itg2mono.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)
81 i1fmbf 23656 . . . . . . . . . . 11 (((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ dom ∫1 → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ MblFn)
8220, 81syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((ℝ × {-𝑇}) ∘𝑓 · 𝐻) ∈ MblFn)
8380, 82mbfadd 23642 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) ∈ MblFn)
84 mbfima 23611 . . . . . . . . 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)
8583, 26, 84syl2anc 575 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ∈ dom vol)
86 cmmbl 23515 . . . . . . . 8 ((((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)) ∈ dom vol → (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ∈ dom vol)
8785, 86syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0))) ∈ dom vol)
88 inmbl 23523 . . . . . . 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)
8979, 87, 88sylancr 577 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (ℝ ∩ (ℝ ∖ (((𝐹𝑛) ∘𝑓 + ((ℝ × {-𝑇}) ∘𝑓 · 𝐻)) “ (-∞(,)0)))) ∈ dom vol)
9078, 89eqeltrrd 2886 . . . . 5 ((𝜑𝑛 ∈ ℕ) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} ∈ dom vol)
91 itg2mono.11 . . . . 5 𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})
9290, 91fmptd 6606 . . . 4 (𝜑𝐴:ℕ⟶dom vol)
93 itg2mono.4 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
9493ralrimiva 3154 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
95 fveq2 6408 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
96 fvoveq1 6897 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑗 + 1)))
9795, 96breq12d 4857 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)) ↔ (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1))))
9897cbvralv 3360 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)))
9994, 98sylib 209 . . . . . . . . . 10 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)))
10099r19.21bi 3120 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)))
1015ralrimiva 3154 . . . . . . . . . . . . 13 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞))
10295feq1d 6241 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑗):ℝ⟶(0[,)+∞)))
103102cbvralv 3360 . . . . . . . . . . . . 13 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
104101, 103sylib 209 . . . . . . . . . . . 12 (𝜑 → ∀𝑗 ∈ ℕ (𝐹𝑗):ℝ⟶(0[,)+∞))
105104r19.21bi 3120 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
106105ffnd 6257 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) Fn ℝ)
107 peano2nn 11317 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
108 fveq2 6408 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (𝐹𝑛) = (𝐹‘(𝑗 + 1)))
109108feq1d 6241 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞)))
110109rspccva 3501 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (𝑗 + 1) ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
111101, 107, 110syl2an 585 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞))
112111ffnd 6257 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) Fn ℝ)
11323a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ℝ ∈ V)
114 eqidd 2807 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑥))
115 eqidd 2807 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
116106, 112, 113, 113, 25, 114, 115ofrfval 7135 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((𝐹𝑗) ∘𝑟 ≤ (𝐹‘(𝑗 + 1)) ↔ ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
117100, 116mpbid 223 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
118117r19.21bi 3120 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥))
11917ad2antrr 708 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
12044adantr 468 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 𝐻:ℝ⟶ℝ)
121120ffvelrnda 6581 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐻𝑥) ∈ ℝ)
122119, 121remulcld 10355 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
123 fss 6269 . . . . . . . . . 10 (((𝐹𝑗):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹𝑗):ℝ⟶ℝ)
124105, 6, 123sylancl 576 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶ℝ)
125124ffvelrnda 6581 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
126 fss 6269 . . . . . . . . . 10 (((𝐹‘(𝑗 + 1)):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
127111, 6, 126sylancl 576 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)):ℝ⟶ℝ)
128127ffvelrnda 6581 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑗 + 1))‘𝑥) ∈ ℝ)
129 letr 10416 . . . . . . . 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)))
130122, 125, 128, 129syl3anc 1483 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) ∧ ((𝐹𝑗)‘𝑥) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
131118, 130mpan2d 677 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
132131ss2rabdv 3880 . . . . 5 ((𝜑𝑗 ∈ ℕ) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ⊆ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
13395fveq1d 6410 . . . . . . . . 9 (𝑛 = 𝑗 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑗)‘𝑥))
134133breq2d 4856 . . . . . . . 8 (𝑛 = 𝑗 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
135134rabbidv 3379 . . . . . . 7 (𝑛 = 𝑗 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
13623rabex 5007 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ∈ V
137135, 91, 136fvmpt 6503 . . . . . 6 (𝑗 ∈ ℕ → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
138137adantl 469 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
139107adantl 469 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
140108fveq1d 6410 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐹𝑛)‘𝑥) = ((𝐹‘(𝑗 + 1))‘𝑥))
141140breq2d 4856 . . . . . . . 8 (𝑛 = (𝑗 + 1) → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)))
142141rabbidv 3379 . . . . . . 7 (𝑛 = (𝑗 + 1) → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
14323rabex 5007 . . . . . . 7 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)} ∈ V
144142, 91, 143fvmpt 6503 . . . . . 6 ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
145139, 144syl 17 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐴‘(𝑗 + 1)) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹‘(𝑗 + 1))‘𝑥)})
146132, 138, 1453sstr4d 3845 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ⊆ (𝐴‘(𝑗 + 1)))
14760adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
14851adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℝ)
14957an32s 634 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
150149fmpttd 6607 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)):ℕ⟶ℝ)
151150frnd 6263 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
152 1nn 11316 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
153 eqid 2806 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))
154153, 149dmmptd 6235 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ℕ)
155152, 154syl5eleqr 2892 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)))
156155ne0d 4123 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
157 dm0rn0 5543 . . . . . . . . . . . . . . . . . 18 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ∅)
158157necon3bii 3030 . . . . . . . . . . . . . . . . 17 (dom (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
159156, 158sylib 209 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
160 itg2mono.5 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
161150ffnd 6257 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
162 breq1 4847 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) → (𝑧𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
163162ralrn 6584 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
164161, 163syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
165 fveq2 6408 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
166165fveq1d 6410 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑚)‘𝑥))
167 fvex 6421 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑚)‘𝑥) ∈ V
168166, 153, 167fvmpt 6503 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) = ((𝐹𝑚)‘𝑥))
169168breq1d 4854 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
170169ralbiia 3167 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
171166breq1d 4854 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹𝑚)‘𝑥) ≤ 𝑦))
172171cbvralv 3360 . . . . . . . . . . . . . . . . . . . 20 (∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹𝑚)‘𝑥) ≤ 𝑦)
173170, 172bitr4i 269 . . . . . . . . . . . . . . . . . . 19 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)
174164, 173syl6bb 278 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
175174rexbidv 3240 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦))
176160, 175mpbird 248 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
177 suprcl 11268 . . . . . . . . . . . . . . . 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)
178151, 159, 176, 177syl3anc 1483 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
179178adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
18016simp3d 1167 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 < 1)
181180adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 < 1)
18217adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 𝑇 ∈ ℝ)
183 1red 10326 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 1 ∈ ℝ)
184 simprr 780 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → 0 < (𝐻𝑥))
185 ltmul1 11158 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝐻𝑥) ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
186182, 183, 148, 184, 185syl112anc 1486 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 < 1 ↔ (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥))))
187181, 186mpbid 223 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < (1 · (𝐻𝑥)))
188148recnd 10353 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ∈ ℂ)
189188mulid2d 10343 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (1 · (𝐻𝑥)) = (𝐻𝑥))
190187, 189breqtrd 4870 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < (𝐻𝑥))
191 itg2mono.9 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻𝑟𝐺)
192 itg2mono.1 . . . . . . . . . . . . . . . . . . . . 21 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
193178, 192fmptd 6606 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:ℝ⟶ℝ)
194193ffnd 6257 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 Fn ℝ)
19523a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ℝ ∈ V)
196 eqidd 2807 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐻𝑦) = (𝐻𝑦))
197 fveq2 6408 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑦 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑦))
198197mpteq2dv 4939 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
199198rneqd 5554 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) = ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)))
200199supeq1d 8591 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
201 ltso 10403 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
202201supex 8608 . . . . . . . . . . . . . . . . . . . . 21 sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ) ∈ V
203200, 192, 202fvmpt 6503 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℝ → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
204203adantl 469 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ ℝ) → (𝐺𝑦) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
20545, 194, 195, 195, 25, 196, 204ofrfval 7135 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐻𝑟𝐺 ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
206191, 205mpbid 223 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
207 fveq2 6408 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐻𝑥) = (𝐻𝑦))
208207, 200breq12d 4857 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < )))
209208cbvralv 3360 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∀𝑦 ∈ ℝ (𝐻𝑦) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑦)), ℝ, < ))
210206, 209sylibr 225 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ ℝ (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
211210r19.21bi 3120 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ℝ) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
212211adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝐻𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
213147, 148, 179, 190, 212ltletrd 10482 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
214151adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ⊆ ℝ)
215159adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) ≠ ∅)
216176adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))𝑧𝑦)
217 suprlub 11272 . . . . . . . . . . . . . 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))
218214, 215, 216, 147, 217syl31anc 1485 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ((𝑇 · (𝐻𝑥)) < sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) ↔ ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤))
219213, 218mpbid 223 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤)
220161adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)) Fn ℕ)
221 breq2 4848 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) → ((𝑇 · (𝐻𝑥)) < 𝑤 ↔ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗)))
222221rexrn 6583 . . . . . . . . . . . . . 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)))
223220, 222syl 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)))
224 fvex 6421 . . . . . . . . . . . . . . . 16 ((𝐹𝑗)‘𝑥) ∈ V
225133, 153, 224fvmpt 6503 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) = ((𝐹𝑗)‘𝑥))
226225breq2d 4856 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ((𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
227226rexbiia 3228 . . . . . . . . . . . . 13 (∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))‘𝑗) ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
228223, 227syl6bb 278 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥))(𝑇 · (𝐻𝑥)) < 𝑤 ↔ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥)))
229219, 228mpbid 223 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥))
230182, 148remulcld 10355 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
231230adantr 468 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
232105adantlr 697 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗):ℝ⟶(0[,)+∞))
233 simplr 776 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑥 ∈ ℝ)
234232, 233ffvelrnd 6582 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ (0[,)+∞))
235 elrege0 12498 . . . . . . . . . . . . . . . 16 (((𝐹𝑗)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
236234, 235sylib 209 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
237236simpld 484 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
238237adantlrr 703 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
239 ltle 10411 . . . . . . . . . . . . 13 (((𝑇 · (𝐻𝑥)) ∈ ℝ ∧ ((𝐹𝑗)‘𝑥) ∈ ℝ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
240231, 238, 239syl2anc 575 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) ∧ 𝑗 ∈ ℕ) → ((𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
241240reximdva 3204 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → (∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) < ((𝐹𝑗)‘𝑥) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)))
242229, 241mpd 15 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 < (𝐻𝑥))) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
243242anassrs 455 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 0 < (𝐻𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
244152ne0ii 4125 . . . . . . . . . . 11 ℕ ≠ ∅
24560adantrr 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
246245adantr 468 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ∈ ℝ)
247 0red 10328 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ)
248236adantlrr 703 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (((𝐹𝑗)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑗)‘𝑥)))
249248simpld 484 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗)‘𝑥) ∈ ℝ)
250 simplrr 787 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ≤ 0)
25151adantrr 699 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → (𝐻𝑥) ∈ ℝ)
252251adantr 468 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝐻𝑥) ∈ ℝ)
25317ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
25416simp2d 1166 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑇)
255254ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 < 𝑇)
256 lemul2 11161 . . . . . . . . . . . . . . . 16 (((𝐻𝑥) ∈ ℝ ∧ 0 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
257252, 247, 253, 255, 256syl112anc 1486 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → ((𝐻𝑥) ≤ 0 ↔ (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0)))
258250, 257mpbid 223 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ (𝑇 · 0))
259253recnd 10353 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 𝑇 ∈ ℂ)
260259mul01d 10520 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · 0) = 0)
261258, 260breqtrd 4870 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ 0)
262248simprd 485 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → 0 ≤ ((𝐹𝑗)‘𝑥))
263246, 247, 249, 261, 262letrd 10479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) ∧ 𝑗 ∈ ℕ) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
264263ralrimiva 3154 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
265 r19.2z 4255 . . . . . . . . . . 11 ((ℕ ≠ ∅ ∧ ∀𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
266244, 264, 265sylancr 577 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐻𝑥) ≤ 0)) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
267266anassrs 455 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝐻𝑥) ≤ 0) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
268 0red 10328 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 0 ∈ ℝ)
269243, 267, 268, 51ltlecasei 10430 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
270269ralrimiva 3154 . . . . . . 7 (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
271 rabid2 3307 . . . . . . 7 (ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥))
272270, 271sylibr 225 . . . . . 6 (𝜑 → ℝ = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
273 iunrab 4759 . . . . . 6 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)} = {𝑥 ∈ ℝ ∣ ∃𝑗 ∈ ℕ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)}
274272, 273syl6eqr 2858 . . . . 5 (𝜑 → ℝ = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
275138iuneq2dv 4734 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = 𝑗 ∈ ℕ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑗)‘𝑥)})
27692ffnd 6257 . . . . . 6 (𝜑𝐴 Fn ℕ)
277 fniunfv 6729 . . . . . 6 (𝐴 Fn ℕ → 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
278276, 277syl 17 . . . . 5 (𝜑 𝑗 ∈ ℕ (𝐴𝑗) = ran 𝐴)
279274, 275, 2783eqtr2rd 2847 . . . 4 (𝜑 ran 𝐴 = ℝ)
280 eqid 2806 . . . 4 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))
28192, 146, 279, 9, 280itg1climres 23695 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ⇝ (∫1𝐻))
282 nnex 11311 . . . . 5 ℕ ∈ V
283282mptex 6711 . . . 4 (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ∈ V
284283a1i 11 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ∈ V)
285 fveq2 6408 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
286285eleq2d 2871 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑥 ∈ (𝐴𝑗) ↔ 𝑥 ∈ (𝐴𝑘)))
287286ifbid 4301 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0) = if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
288287mpteq2dv 4939 . . . . . . . 8 (𝑗 = 𝑘 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
289288fveq2d 6412 . . . . . . 7 (𝑗 = 𝑘 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
290 eqid 2806 . . . . . . 7 (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))
291 fvex 6421 . . . . . . 7 (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ V
292289, 290, 291fvmpt 6503 . . . . . 6 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
293292adantl 469 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
2949adantr 468 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → 𝐻 ∈ dom ∫1)
29592ffvelrnda 6581 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐴𝑘) ∈ dom vol)
296 eqid 2806 . . . . . . . 8 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))
297296i1fres 23686 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑘) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
298294, 295, 297syl2anc 575 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1)
299 itg1cl 23666 . . . . . 6 ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) ∈ dom ∫1 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ ℝ)
300298, 299syl 17 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ ℝ)
301293, 300eqeltrd 2885 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℝ)
302301recnd 10353 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘) ∈ ℂ)
303289oveq2d 6890 . . . . . 6 (𝑗 = 𝑘 → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
304 eqid 2806 . . . . . 6 (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) = (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))
305 ovex 6906 . . . . . 6 (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) ∈ V
306303, 304, 305fvmpt 6503 . . . . 5 (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
307292oveq2d 6890 . . . . 5 (𝑘 ∈ ℕ → (𝑇 · ((𝑗 ∈ ℕ ↦ (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))‘𝑘)) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
308306, 307eqtr4d 2843 . . . 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)))
309308adantl 469 . . 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)))
3101, 2, 281, 49, 284, 302, 309climmulc2 14590 . 2 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))) ⇝ (𝑇 · (∫1𝐻)))
311 icossicc 12479 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
312 fss 6269 . . . . . . 7 (((𝐹𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑛):ℝ⟶(0[,]+∞))
3135, 311, 312sylancl 576 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,]+∞))
314 itg2mono.10 . . . . . . 7 (𝜑𝑆 ∈ ℝ)
315314adantr 468 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
316 itg2cl 23713 . . . . . . . . . . . 12 ((𝐹𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
317313, 316syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ*)
318317fmpttd 6607 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ*)
319318frnd 6263 . . . . . . . . 9 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
320319adantr 468 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ*)
321 fvex 6421 . . . . . . . . . . 11 (∫2‘(𝐹𝑛)) ∈ V
322321elabrex 6725 . . . . . . . . . 10 (𝑛 ∈ ℕ → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
323322adantl 469 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))})
324 eqid 2806 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))
325324rnmpt 5572 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (∫2‘(𝐹𝑛))}
326323, 325syl6eleqr 2896 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))))
327 supxrub 12372 . . . . . . . 8 ((ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ* ∧ (∫2‘(𝐹𝑛)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))) → (∫2‘(𝐹𝑛)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ))
328320, 326, 327syl2anc 575 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ))
329 itg2mono.6 . . . . . . 7 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )
330328, 329syl6breqr 4886 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ 𝑆)
331 itg2lecl 23719 . . . . . 6 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ ∧ (∫2‘(𝐹𝑛)) ≤ 𝑆) → (∫2‘(𝐹𝑛)) ∈ ℝ)
332313, 315, 330, 331syl3anc 1483 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ∈ ℝ)
333332fmpttd 6607 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))):ℕ⟶ℝ)
334313ralrimiva 3154 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞))
335 fveq2 6408 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
336335feq1d 6241 . . . . . . . . . . 11 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹𝑘):ℝ⟶(0[,]+∞)))
337336cbvralv 3360 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,]+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
338334, 337sylib 209 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞))
339 peano2nn 11317 . . . . . . . . 9 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
340 fveq2 6408 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
341340feq1d 6241 . . . . . . . . . 10 (𝑘 = (𝑛 + 1) → ((𝐹𝑘):ℝ⟶(0[,]+∞) ↔ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞)))
342341rspccva 3501 . . . . . . . . 9 ((∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,]+∞) ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
343338, 339, 342syl2an 585 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞))
344 itg2le 23720 . . . . . . . 8 (((𝐹𝑛):ℝ⟶(0[,]+∞) ∧ (𝐹‘(𝑛 + 1)):ℝ⟶(0[,]+∞) ∧ (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1))) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
345313, 343, 93, 344syl3anc 1483 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
346345ralrimiva 3154 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
347 2fveq3 6413 . . . . . . . . . 10 (𝑛 = 𝑘 → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹𝑘)))
348 fvex 6421 . . . . . . . . . 10 (∫2‘(𝐹𝑘)) ∈ V
349347, 324, 348fvmpt 6503 . . . . . . . . 9 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
350 peano2nn 11317 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
351 2fveq3 6413 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (∫2‘(𝐹𝑛)) = (∫2‘(𝐹‘(𝑘 + 1))))
352 fvex 6421 . . . . . . . . . . 11 (∫2‘(𝐹‘(𝑘 + 1))) ∈ V
353351, 324, 352fvmpt 6503 . . . . . . . . . 10 ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) = (∫2‘(𝐹‘(𝑘 + 1))))
354350, 353syl 17 . . . . . . . . 9 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) = (∫2‘(𝐹‘(𝑘 + 1))))
355349, 354breq12d 4857 . . . . . . . 8 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
356355ralbiia 3167 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
357 fvoveq1 6897 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
358357fveq2d 6412 . . . . . . . . 9 (𝑛 = 𝑘 → (∫2‘(𝐹‘(𝑛 + 1))) = (∫2‘(𝐹‘(𝑘 + 1))))
359347, 358breq12d 4857 . . . . . . . 8 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1)))))
360359cbvralv 3360 . . . . . . 7 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))) ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ (∫2‘(𝐹‘(𝑘 + 1))))
361356, 360bitr4i 269 . . . . . 6 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)) ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ (∫2‘(𝐹‘(𝑛 + 1))))
362346, 361sylibr 225 . . . . 5 (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
363362r19.21bi 3120 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘(𝑘 + 1)))
364330ralrimiva 3154 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆)
365349breq1d 4854 . . . . . . . . 9 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
366365ralbiia 3167 . . . . . . . 8 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
367347breq1d 4854 . . . . . . . . 9 (𝑛 = 𝑘 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑘)) ≤ 𝑥))
368367cbvralv 3360 . . . . . . . 8 (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (∫2‘(𝐹𝑘)) ≤ 𝑥)
369366, 368bitr4i 269 . . . . . . 7 (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥)
370 breq2 4848 . . . . . . . 8 (𝑥 = 𝑆 → ((∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ (∫2‘(𝐹𝑛)) ≤ 𝑆))
371370ralbidv 3174 . . . . . . 7 (𝑥 = 𝑆 → (∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
372369, 371syl5bb 274 . . . . . 6 (𝑥 = 𝑆 → (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥 ↔ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆))
373372rspcev 3502 . . . . 5 ((𝑆 ∈ ℝ ∧ ∀𝑛 ∈ ℕ (∫2‘(𝐹𝑛)) ≤ 𝑆) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
374314, 364, 373syl2anc 575 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥)
3751, 2, 333, 363, 374climsup 14623 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
376333frnd 6263 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⊆ ℝ)
377324, 317dmmptd 6235 . . . . . . 7 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ℕ)
378244a1i 11 . . . . . . 7 (𝜑 → ℕ ≠ ∅)
379377, 378eqnetrd 3045 . . . . . 6 (𝜑 → dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
380 dm0rn0 5543 . . . . . . 7 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) = ∅)
381380necon3bii 3030 . . . . . 6 (dom (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
382379, 381sylib 209 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ≠ ∅)
383321, 324fnmpti 6233 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ
384383a1i 11 . . . . . . . 8 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ)
385 breq1 4847 . . . . . . . . 9 (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) → (𝑧𝑥 ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
386385ralrn 6584 . . . . . . . 8 ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
387384, 386syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
388387rexbidv 3240 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ≤ 𝑥))
389374, 388mpbird 248 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))𝑧𝑥)
390 supxrre 12375 . . . . 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, < ))
391376, 382, 389, 390syl3anc 1483 . . . 4 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ))
392329, 391syl5req 2853 . . 3 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ, < ) = 𝑆)
393375, 392breqtrd 4870 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))) ⇝ 𝑆)
39417adantr 468 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝑇 ∈ ℝ)
3959adantr 468 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝐻 ∈ dom ∫1)
39692ffvelrnda 6581 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐴𝑗) ∈ dom vol)
397280i1fres 23686 . . . . . . 7 ((𝐻 ∈ dom ∫1 ∧ (𝐴𝑗) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
398395, 396, 397syl2anc 575 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1)
399 itg1cl 23666 . . . . . 6 ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)) ∈ dom ∫1 → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) ∈ ℝ)
400398, 399syl 17 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))) ∈ ℝ)
401394, 400remulcld 10355 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))) ∈ ℝ)
402401fmpttd 6607 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0))))):ℕ⟶ℝ)
403402ffvelrnda 6581 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) ∈ ℝ)
404333ffvelrnda 6581 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) ∈ ℝ)
405335feq1d 6241 . . . . . . . 8 (𝑛 = 𝑘 → ((𝐹𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹𝑘):ℝ⟶(0[,)+∞)))
406405cbvralv 3360 . . . . . . 7 (∀𝑛 ∈ ℕ (𝐹𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
407101, 406sylib 209 . . . . . 6 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘):ℝ⟶(0[,)+∞))
408407r19.21bi 3120 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,)+∞))
409 fss 6269 . . . . 5 (((𝐹𝑘):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹𝑘):ℝ⟶(0[,]+∞))
410408, 311, 409sylancl 576 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘):ℝ⟶(0[,]+∞))
41123a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ℝ ∈ V)
41217adantr 468 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℝ)
413412adantr 468 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
414 fvex 6421 . . . . . . . . 9 (𝐻𝑥) ∈ V
415 c0ex 10319 . . . . . . . . 9 0 ∈ V
416414, 415ifex 4327 . . . . . . . 8 if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V
417416a1i 11 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0) ∈ V)
418 fconstmpt 5363 . . . . . . . 8 (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇)
419418a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (ℝ × {𝑇}) = (𝑥 ∈ ℝ ↦ 𝑇))
420 eqidd 2807 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))
421411, 413, 417, 419, 420offval2 7144 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))))
422 ovif2 6968 . . . . . . . 8 (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0))
42349adantr 468 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑇 ∈ ℂ)
424423mul01d 10520 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝑇 · 0) = 0)
425424ifeq2d 4298 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), (𝑇 · 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
426422, 425syl5eq 2852 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)) = if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))
427426mpteq2dv 4939 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ (𝑇 · if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
428421, 427eqtrd 2840 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
429298, 412i1fmulc 23684 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0))) ∈ dom ∫1)
430428, 429eqeltrrd 2886 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∈ dom ∫1)
431 iftrue 4285 . . . . . . . . 9 (𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
432431adantl 469 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = (𝑇 · (𝐻𝑥)))
433335fveq1d 6410 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑘)‘𝑥))
434433breq2d 4856 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → ((𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥) ↔ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
435434rabbidv 3379 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
43623rabex 5007 . . . . . . . . . . . . 13 {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ∈ V
437435, 91, 436fvmpt 6503 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
438437ad2antlr 709 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝑘) = {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
439438eleq2d 2871 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐴𝑘) ↔ 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)}))
440439biimpa 464 . . . . . . . . 9 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → 𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)})
441 rabid 3304 . . . . . . . . . 10 (𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} ↔ (𝑥 ∈ ℝ ∧ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)))
442441simprbi 486 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥)} → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥))
443440, 442syl 17 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑘)‘𝑥))
444432, 443eqbrtrd 4866 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
445 iffalse 4288 . . . . . . . . 9 𝑥 ∈ (𝐴𝑘) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
446445adantl 469 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) = 0)
447408ffvelrnda 6581 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ (0[,)+∞))
448 elrege0 12498 . . . . . . . . . . 11 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹𝑘)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹𝑘)‘𝑥)))
449448simprbi 486 . . . . . . . . . 10 (((𝐹𝑘)‘𝑥) ∈ (0[,)+∞) → 0 ≤ ((𝐹𝑘)‘𝑥))
450447, 449syl 17 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹𝑘)‘𝑥))
451450adantr 468 . . . . . . . 8 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → 0 ≤ ((𝐹𝑘)‘𝑥))
452446, 451eqbrtrd 4866 . . . . . . 7 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (𝐴𝑘)) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
453444, 452pm2.61dan 838 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
454453ralrimiva 3154 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥))
455 ovex 6906 . . . . . . . 8 (𝑇 · (𝐻𝑥)) ∈ V
456455, 415ifex 4327 . . . . . . 7 if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V
457456a1i 11 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ∈ V)
458 fvexd 6423 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑘)‘𝑥) ∈ V)
459 eqidd 2807 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)))
460408feqmptd 6470 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = (𝑥 ∈ ℝ ↦ ((𝐹𝑘)‘𝑥)))
461411, 457, 458, 459, 460ofrfval2 7145 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘𝑟 ≤ (𝐹𝑘) ↔ ∀𝑥 ∈ ℝ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0) ≤ ((𝐹𝑘)‘𝑥)))
462454, 461mpbird 248 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0)) ∘𝑟 ≤ (𝐹𝑘))
463 itg2ub 23714 . . . 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)))
464410, 430, 462, 463syl3anc 1483 . . 3 ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))) ≤ (∫2‘(𝐹𝑘)))
465306adantl 469 . . . 4 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
466298, 412itg1mulc 23685 . . . 4 ((𝜑𝑘 ∈ ℕ) → (∫1‘((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) = (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))))
467428fveq2d 6412 . . . 4 ((𝜑𝑘 ∈ ℕ) → (∫1‘((ℝ × {𝑇}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝐻𝑥), 0)))) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))))
468465, 466, 4673eqtr2d 2846 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) = (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑘), (𝑇 · (𝐻𝑥)), 0))))
469349adantl 469 . . 3 ((𝜑𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘) = (∫2‘(𝐹𝑘)))
470464, 468, 4693brtr4d 4876 . 2 ((𝜑𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑇 · (∫1‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑗), (𝐻𝑥), 0)))))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛)))‘𝑘))
4711, 2, 310, 393, 403, 404, 470climle 14593 1 (𝜑 → (𝑇 · (∫1𝐻)) ≤ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  {crab 3100  Vcvv 3391  cdif 3766  cin 3768  wss 3769  c0 4116  ifcif 4279  {csn 4370   cuni 4630   ciun 4712   class class class wbr 4844  cmpt 4923   × cxp 5309  ccnv 5310  dom cdm 5311  ran crn 5312  cima 5314   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6874  𝑓 cof 7125  𝑟 cofr 7126  supcsup 8585  cc 10219  cr 10220  0cc0 10221  1c1 10222   + caddc 10224   · cmul 10226  +∞cpnf 10356  -∞cmnf 10357  *cxr 10358   < clt 10359  cle 10360  cmin 10551  -cneg 10552  cn 11305  (,)cioo 12393  [,)cico 12395  [,]cicc 12396  cli 14438  volcvol 23444  MblFncmbf 23595  1citg1 23596  2citg2 23597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cc 9542  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299  ax-addf 10300
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-disj 4813  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-ofr 7128  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-omul 7801  df-er 7979  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fi 8556  df-sup 8587  df-inf 8588  df-oi 8654  df-card 9048  df-acn 9051  df-cda 9275  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-n0 11560  df-z 11644  df-uz 11905  df-q 12008  df-rp 12047  df-xneg 12162  df-xadd 12163  df-xmul 12164  df-ioo 12397  df-ioc 12398  df-ico 12399  df-icc 12400  df-fz 12550  df-fzo 12690  df-fl 12817  df-seq 13025  df-exp 13084  df-hash 13338  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-clim 14442  df-rlim 14443  df-sum 14640  df-rest 16288  df-topgen 16309  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-top 20912  df-topon 20929  df-bases 20964  df-cmp 21404  df-ovol 23445  df-vol 23446  df-mbf 23600  df-itg1 23601  df-itg2 23602
This theorem is referenced by:  itg2monolem3  23733
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