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Theorem mbfi1fseqlem6 23532
Description: Lemma for mbfi1fseq 23533. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existence quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1 (𝜑𝐹 ∈ MblFn)
mbfi1fseq.2 (𝜑𝐹:ℝ⟶(0[,)+∞))
mbfi1fseq.3 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))
mbfi1fseq.4 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
Assertion
Ref Expression
mbfi1fseqlem6 (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
Distinct variable groups:   𝑔,𝑚,𝑛,𝑥,𝑦,𝐹   𝑔,𝐺,𝑛,𝑥   𝑚,𝐽   𝜑,𝑚,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑔)   𝐺(𝑦,𝑚)   𝐽(𝑥,𝑦,𝑔,𝑛)

Proof of Theorem mbfi1fseqlem6
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfi1fseq.1 . . 3 (𝜑𝐹 ∈ MblFn)
2 mbfi1fseq.2 . . 3 (𝜑𝐹:ℝ⟶(0[,)+∞))
3 mbfi1fseq.3 . . 3 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))
4 mbfi1fseq.4 . . 3 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
51, 2, 3, 4mbfi1fseqlem4 23530 . 2 (𝜑𝐺:ℕ⟶dom ∫1)
61, 2, 3, 4mbfi1fseqlem5 23531 . . 3 ((𝜑𝑛 ∈ ℕ) → (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
76ralrimiva 2995 . 2 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
8 simpr 476 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
98recnd 10106 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
109abscld 14219 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (abs‘𝑥) ∈ ℝ)
112ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
12 elrege0 12316 . . . . . . . 8 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1311, 12sylib 208 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1413simpld 474 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
1510, 14readdcld 10107 . . . . 5 ((𝜑𝑥 ∈ ℝ) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
16 arch 11327 . . . . 5 (((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ → ∃𝑘 ∈ ℕ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)
1715, 16syl 17 . . . 4 ((𝜑𝑥 ∈ ℝ) → ∃𝑘 ∈ ℕ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)
18 eqid 2651 . . . . 5 (ℤ𝑘) = (ℤ𝑘)
19 nnz 11437 . . . . . 6 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
2019ad2antrl 764 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℤ)
21 nnuz 11761 . . . . . . . 8 ℕ = (ℤ‘1)
22 1zzd 11446 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 1 ∈ ℤ)
23 halfcn 11285 . . . . . . . . . 10 (1 / 2) ∈ ℂ
2423a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (1 / 2) ∈ ℂ)
25 halfre 11284 . . . . . . . . . . . 12 (1 / 2) ∈ ℝ
26 0re 10078 . . . . . . . . . . . . 13 0 ∈ ℝ
27 halfgt0 11286 . . . . . . . . . . . . 13 0 < (1 / 2)
2826, 25, 27ltleii 10198 . . . . . . . . . . . 12 0 ≤ (1 / 2)
29 absid 14080 . . . . . . . . . . . 12 (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2))
3025, 28, 29mp2an 708 . . . . . . . . . . 11 (abs‘(1 / 2)) = (1 / 2)
31 halflt1 11288 . . . . . . . . . . 11 (1 / 2) < 1
3230, 31eqbrtri 4706 . . . . . . . . . 10 (abs‘(1 / 2)) < 1
3332a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (abs‘(1 / 2)) < 1)
3424, 33expcnv 14640 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) ⇝ 0)
3514recnd 10106 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℂ)
36 nnex 11064 . . . . . . . . . 10 ℕ ∈ V
3736mptex 6527 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V
3837a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V)
39 nnnn0 11337 . . . . . . . . . . 11 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
4039adantl 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
41 oveq2 6698 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((1 / 2)↑𝑛) = ((1 / 2)↑𝑗))
42 eqid 2651 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))
43 ovex 6718 . . . . . . . . . . 11 ((1 / 2)↑𝑗) ∈ V
4441, 42, 43fvmpt 6321 . . . . . . . . . 10 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) = ((1 / 2)↑𝑗))
4540, 44syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) = ((1 / 2)↑𝑗))
46 expcl 12918 . . . . . . . . . 10 (((1 / 2) ∈ ℂ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℂ)
4723, 40, 46sylancr 696 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((1 / 2)↑𝑗) ∈ ℂ)
4845, 47eqeltrd 2730 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) ∈ ℂ)
4941oveq2d 6706 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝐹𝑥) − ((1 / 2)↑𝑛)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
50 eqid 2651 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))
51 ovex 6718 . . . . . . . . . . 11 ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ V
5249, 50, 51fvmpt 6321 . . . . . . . . . 10 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5352adantl 481 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5445oveq2d 6706 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5553, 54eqtr4d 2688 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)))
5621, 22, 34, 35, 38, 48, 55climsubc2 14413 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ ((𝐹𝑥) − 0))
5735subid1d 10419 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) − 0) = (𝐹𝑥))
5856, 57breqtrd 4711 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
5958adantr 480 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
6036mptex 6527 . . . . . 6 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V
6160a1i 11 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V)
62 simprl 809 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℕ)
63 eluznn 11796 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ)
6462, 63sylan 487 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ)
6564, 52syl 17 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
6614ad2antrr 762 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ ℝ)
6764, 39syl 17 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ0)
68 reexpcl 12917 . . . . . . . 8 (((1 / 2) ∈ ℝ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℝ)
6925, 67, 68sylancr 696 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) ∈ ℝ)
7066, 69resubcld 10496 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ ℝ)
7165, 70eqeltrd 2730 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ∈ ℝ)
72 fveq2 6229 . . . . . . . . 9 (𝑛 = 𝑗 → (𝐺𝑛) = (𝐺𝑗))
7372fveq1d 6231 . . . . . . . 8 (𝑛 = 𝑗 → ((𝐺𝑛)‘𝑥) = ((𝐺𝑗)‘𝑥))
74 eqid 2651 . . . . . . . 8 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))
75 fvex 6239 . . . . . . . 8 ((𝐺𝑗)‘𝑥) ∈ V
7673, 74, 75fvmpt 6321 . . . . . . 7 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((𝐺𝑗)‘𝑥))
7764, 76syl 17 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((𝐺𝑗)‘𝑥))
785ad3antrrr 766 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐺:ℕ⟶dom ∫1)
7978, 64ffvelrnd 6400 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐺𝑗) ∈ dom ∫1)
80 i1ff 23488 . . . . . . . 8 ((𝐺𝑗) ∈ dom ∫1 → (𝐺𝑗):ℝ⟶ℝ)
8179, 80syl 17 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐺𝑗):ℝ⟶ℝ)
828ad2antrr 762 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
8381, 82ffvelrnd 6400 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐺𝑗)‘𝑥) ∈ ℝ)
8477, 83eqeltrd 2730 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) ∈ ℝ)
8535ad2antrr 762 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ ℂ)
86 2nn 11223 . . . . . . . . . . . . . 14 2 ∈ ℕ
87 nnexpcl 12913 . . . . . . . . . . . . . 14 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
8886, 67, 87sylancr 696 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℕ)
8988nnred 11073 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℝ)
9089recnd 10106 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℂ)
9188nnne0d 11103 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ≠ 0)
9285, 90, 91divcan4d 10845 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)) = (𝐹𝑥))
9392eqcomd 2657 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) = (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)))
94 2cnd 11131 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ∈ ℂ)
95 2ne0 11151 . . . . . . . . . . 11 2 ≠ 0
9695a1i 11 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ≠ 0)
97 eluzelz 11735 . . . . . . . . . . 11 (𝑗 ∈ (ℤ𝑘) → 𝑗 ∈ ℤ)
9897adantl 481 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℤ)
9994, 96, 98exprecd 13056 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
10093, 99oveq12d 6708 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)) − (1 / (2↑𝑗))))
10166, 89remulcld 10108 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℝ)
102101recnd 10106 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℂ)
103 1cnd 10094 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℂ)
104102, 103, 90, 91divsubdird 10878 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) = ((((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)) − (1 / (2↑𝑗))))
105100, 104eqtr4d 2688 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)))
106 fllep1 12642 . . . . . . . . . 10 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → ((𝐹𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) + 1))
107101, 106syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) + 1))
108 1red 10093 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℝ)
109 reflcl 12637 . . . . . . . . . . 11 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → (⌊‘((𝐹𝑥) · (2↑𝑗))) ∈ ℝ)
110101, 109syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (⌊‘((𝐹𝑥) · (2↑𝑗))) ∈ ℝ)
111101, 108, 110lesubaddd 10662 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((((𝐹𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹𝑥) · (2↑𝑗))) ↔ ((𝐹𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) + 1)))
112107, 111mpbird 247 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹𝑥) · (2↑𝑗))))
113 peano2rem 10386 . . . . . . . . . 10 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → (((𝐹𝑥) · (2↑𝑗)) − 1) ∈ ℝ)
114101, 113syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) − 1) ∈ ℝ)
11588nngt0d 11102 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 < (2↑𝑗))
116 lediv1 10926 . . . . . . . . 9 (((((𝐹𝑥) · (2↑𝑗)) − 1) ∈ ℝ ∧ (⌊‘((𝐹𝑥) · (2↑𝑗))) ∈ ℝ ∧ ((2↑𝑗) ∈ ℝ ∧ 0 < (2↑𝑗))) → ((((𝐹𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹𝑥) · (2↑𝑗))) ↔ ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗))))
117114, 110, 89, 115, 116syl112anc 1370 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((((𝐹𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹𝑥) · (2↑𝑗))) ↔ ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗))))
118112, 117mpbid 222 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
119105, 118eqbrtrd 4707 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
1201, 2, 3, 4mbfi1fseqlem2 23528 . . . . . . . . . 10 (𝑗 ∈ ℕ → (𝐺𝑗) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0)))
12164, 120syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐺𝑗) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0)))
122121fveq1d 6231 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐺𝑗)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))‘𝑥))
123 ovex 6718 . . . . . . . . . . 11 (𝑗𝐽𝑥) ∈ V
124 vex 3234 . . . . . . . . . . 11 𝑗 ∈ V
125123, 124ifex 4189 . . . . . . . . . 10 if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) ∈ V
126 c0ex 10072 . . . . . . . . . 10 0 ∈ V
127125, 126ifex 4189 . . . . . . . . 9 if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) ∈ V
128 eqid 2651 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))
129128fvmpt2 6330 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))‘𝑥) = if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))
13082, 127, 129sylancl 695 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))‘𝑥) = if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))
13177, 122, 1303eqtrd 2689 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))
13210ad2antrr 762 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ∈ ℝ)
13315ad2antrr 762 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
13464nnred 11073 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℝ)
13511ad2antrr 762 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ (0[,)+∞))
136135, 12sylib 208 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
137136simprd 478 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (𝐹𝑥))
138132, 66addge01d 10653 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (0 ≤ (𝐹𝑥) ↔ (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥))))
139137, 138mpbid 222 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥)))
14062adantr 480 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℕ)
141140nnred 11073 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℝ)
142 simplrr 818 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)
143133, 141, 142ltled 10223 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑘)
144 eluzle 11738 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑘) → 𝑘𝑗)
145144adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘𝑗)
146133, 141, 134, 143, 145letrd 10232 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑗)
147132, 133, 134, 139, 146letrd 10232 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ≤ 𝑗)
14882, 134absled 14213 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) ≤ 𝑗 ↔ (-𝑗𝑥𝑥𝑗)))
149147, 148mpbid 222 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (-𝑗𝑥𝑥𝑗))
150149simpld 474 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗𝑥)
151149simprd 478 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥𝑗)
152134renegcld 10495 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗 ∈ ℝ)
153 elicc2 12276 . . . . . . . . . 10 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
154152, 134, 153syl2anc 694 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
15582, 150, 151, 154mpbir3and 1264 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ (-𝑗[,]𝑗))
156155iftrued 4127 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) = if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗))
157 simpr 476 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑗𝑦 = 𝑥) → 𝑦 = 𝑥)
158157fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
159 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑗𝑦 = 𝑥) → 𝑚 = 𝑗)
160159oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (2↑𝑚) = (2↑𝑗))
161158, 160oveq12d 6708 . . . . . . . . . . . . . 14 ((𝑚 = 𝑗𝑦 = 𝑥) → ((𝐹𝑦) · (2↑𝑚)) = ((𝐹𝑥) · (2↑𝑗)))
162161fveq2d 6233 . . . . . . . . . . . . 13 ((𝑚 = 𝑗𝑦 = 𝑥) → (⌊‘((𝐹𝑦) · (2↑𝑚))) = (⌊‘((𝐹𝑥) · (2↑𝑗))))
163162, 160oveq12d 6708 . . . . . . . . . . . 12 ((𝑚 = 𝑗𝑦 = 𝑥) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
164 ovex 6718 . . . . . . . . . . . 12 ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ V
165163, 3, 164ovmpt2a 6833 . . . . . . . . . . 11 ((𝑗 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
16664, 82, 165syl2anc 694 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
167110, 88nndivred 11107 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ ℝ)
168 flle 12640 . . . . . . . . . . . . 13 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → (⌊‘((𝐹𝑥) · (2↑𝑗))) ≤ ((𝐹𝑥) · (2↑𝑗)))
169101, 168syl 17 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (⌊‘((𝐹𝑥) · (2↑𝑗))) ≤ ((𝐹𝑥) · (2↑𝑗)))
170 ledivmul2 10940 . . . . . . . . . . . . 13 (((⌊‘((𝐹𝑥) · (2↑𝑗))) ∈ ℝ ∧ (𝐹𝑥) ∈ ℝ ∧ ((2↑𝑗) ∈ ℝ ∧ 0 < (2↑𝑗))) → (((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹𝑥) ↔ (⌊‘((𝐹𝑥) · (2↑𝑗))) ≤ ((𝐹𝑥) · (2↑𝑗))))
171110, 66, 89, 115, 170syl112anc 1370 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹𝑥) ↔ (⌊‘((𝐹𝑥) · (2↑𝑗))) ≤ ((𝐹𝑥) · (2↑𝑗))))
172169, 171mpbird 247 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹𝑥))
1739ad2antrr 762 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℂ)
174173absge0d 14227 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (abs‘𝑥))
17566, 132addge02d 10654 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (0 ≤ (abs‘𝑥) ↔ (𝐹𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥))))
176174, 175mpbid 222 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥)))
17766, 133, 134, 176, 146letrd 10232 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ≤ 𝑗)
178167, 66, 134, 172, 177letrd 10232 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ 𝑗)
179166, 178eqbrtrd 4707 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) ≤ 𝑗)
180179iftrued 4127 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = (𝑗𝐽𝑥))
181180, 166eqtrd 2685 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
182131, 156, 1813eqtrd 2689 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
183119, 65, 1823brtr4d 4717 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗))
184182, 172eqbrtrd 4707 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) ≤ (𝐹𝑥))
18518, 20, 59, 61, 71, 84, 183, 184climsqz 14415 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18617, 185rexlimddv 3064 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
187186ralrimiva 2995 . 2 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18836mptex 6527 . . . 4 (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ∈ V
1894, 188eqeltri 2726 . . 3 𝐺 ∈ V
190 feq1 6064 . . . 4 (𝑔 = 𝐺 → (𝑔:ℕ⟶dom ∫1𝐺:ℕ⟶dom ∫1))
191 fveq1 6228 . . . . . . 7 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
192191breq2d 4697 . . . . . 6 (𝑔 = 𝐺 → (0𝑝𝑟 ≤ (𝑔𝑛) ↔ 0𝑝𝑟 ≤ (𝐺𝑛)))
193 fveq1 6228 . . . . . . 7 (𝑔 = 𝐺 → (𝑔‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
194191, 193breq12d 4698 . . . . . 6 (𝑔 = 𝐺 → ((𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1)) ↔ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
195192, 194anbi12d 747 . . . . 5 (𝑔 = 𝐺 → ((0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
196195ralbidv 3015 . . . 4 (𝑔 = 𝐺 → (∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
197191fveq1d 6231 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛)‘𝑥) = ((𝐺𝑛)‘𝑥))
198197mpteq2dv 4778 . . . . . 6 (𝑔 = 𝐺 → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)))
199198breq1d 4695 . . . . 5 (𝑔 = 𝐺 → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
200199ralbidv 3015 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
201190, 196, 2003anbi123d 1439 . . 3 (𝑔 = 𝐺 → ((𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)) ↔ (𝐺:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
202189, 201spcev 3331 . 2 ((𝐺:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
2035, 7, 187, 202syl3anc 1366 1 (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  ifcif 4119   class class class wbr 4685  cmpt 4762  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑟 cofr 6938  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109   < clt 10112  cle 10113  cmin 10304  -cneg 10305   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  [,)cico 12215  [,]cicc 12216  cfl 12631  cexp 12900  abscabs 14018  cli 14259  MblFncmbf 23428  1citg1 23429  0𝑝c0p 23481
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-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
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-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-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-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-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-0p 23482
This theorem is referenced by:  mbfi1fseq  23533
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