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Theorem mbfi1fseqlem6 23406
Description: Lemma for mbfi1fseq 23407. 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 23404 . 2 (𝜑𝐺:ℕ⟶dom ∫1)
61, 2, 3, 4mbfi1fseqlem5 23405 . . 3 ((𝜑𝑛 ∈ ℕ) → (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
76ralrimiva 2961 . 2 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
8 simpr 477 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
98recnd 10019 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
109abscld 14116 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (abs‘𝑥) ∈ ℝ)
112ffvelrnda 6320 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
12 elrege0 12227 . . . . . . . 8 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1311, 12sylib 208 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1413simpld 475 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
1510, 14readdcld 10020 . . . . 5 ((𝜑𝑥 ∈ ℝ) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
16 arch 11240 . . . . 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 2621 . . . . 5 (ℤ𝑘) = (ℤ𝑘)
19 nnz 11350 . . . . . 6 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
2019ad2antrl 763 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℤ)
21 nnuz 11674 . . . . . . . 8 ℕ = (ℤ‘1)
22 1zzd 11359 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 1 ∈ ℤ)
23 halfcn 11198 . . . . . . . . . 10 (1 / 2) ∈ ℂ
2423a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (1 / 2) ∈ ℂ)
25 halfre 11197 . . . . . . . . . . . 12 (1 / 2) ∈ ℝ
26 0re 9991 . . . . . . . . . . . . 13 0 ∈ ℝ
27 halfgt0 11199 . . . . . . . . . . . . 13 0 < (1 / 2)
2826, 25, 27ltleii 10111 . . . . . . . . . . . 12 0 ≤ (1 / 2)
29 absid 13977 . . . . . . . . . . . 12 (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2))
3025, 28, 29mp2an 707 . . . . . . . . . . 11 (abs‘(1 / 2)) = (1 / 2)
31 halflt1 11201 . . . . . . . . . . 11 (1 / 2) < 1
3230, 31eqbrtri 4639 . . . . . . . . . 10 (abs‘(1 / 2)) < 1
3332a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (abs‘(1 / 2)) < 1)
3424, 33expcnv 14528 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) ⇝ 0)
3514recnd 10019 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℂ)
36 nnex 10977 . . . . . . . . . 10 ℕ ∈ V
3736mptex 6446 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V
3837a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V)
39 nnnn0 11250 . . . . . . . . . . 11 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
4039adantl 482 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
41 oveq2 6618 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((1 / 2)↑𝑛) = ((1 / 2)↑𝑗))
42 eqid 2621 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))
43 ovex 6638 . . . . . . . . . . 11 ((1 / 2)↑𝑗) ∈ V
4441, 42, 43fvmpt 6244 . . . . . . . . . 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 12825 . . . . . . . . . 10 (((1 / 2) ∈ ℂ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℂ)
4723, 40, 46sylancr 694 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((1 / 2)↑𝑗) ∈ ℂ)
4845, 47eqeltrd 2698 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) ∈ ℂ)
4941oveq2d 6626 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝐹𝑥) − ((1 / 2)↑𝑛)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
50 eqid 2621 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))
51 ovex 6638 . . . . . . . . . . 11 ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ V
5249, 50, 51fvmpt 6244 . . . . . . . . . 10 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5352adantl 482 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5445oveq2d 6626 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5553, 54eqtr4d 2658 . . . . . . . 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 14310 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ ((𝐹𝑥) − 0))
5735subid1d 10332 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) − 0) = (𝐹𝑥))
5856, 57breqtrd 4644 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
5958adantr 481 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
6036mptex 6446 . . . . . 6 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V
6160a1i 11 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V)
62 simprl 793 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℕ)
63 eluznn 11709 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ)
6462, 63sylan 488 . . . . . . 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 761 . . . . . . 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 12824 . . . . . . . 8 (((1 / 2) ∈ ℝ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℝ)
6925, 67, 68sylancr 694 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) ∈ ℝ)
7066, 69resubcld 10409 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ ℝ)
7165, 70eqeltrd 2698 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ∈ ℝ)
72 fveq2 6153 . . . . . . . . 9 (𝑛 = 𝑗 → (𝐺𝑛) = (𝐺𝑗))
7372fveq1d 6155 . . . . . . . 8 (𝑛 = 𝑗 → ((𝐺𝑛)‘𝑥) = ((𝐺𝑗)‘𝑥))
74 eqid 2621 . . . . . . . 8 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))
75 fvex 6163 . . . . . . . 8 ((𝐺𝑗)‘𝑥) ∈ V
7673, 74, 75fvmpt 6244 . . . . . . 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 765 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐺:ℕ⟶dom ∫1)
7978, 64ffvelrnd 6321 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐺𝑗) ∈ dom ∫1)
80 i1ff 23362 . . . . . . . 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 761 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
8381, 82ffvelrnd 6321 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐺𝑗)‘𝑥) ∈ ℝ)
8477, 83eqeltrd 2698 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) ∈ ℝ)
8535ad2antrr 761 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ ℂ)
86 2nn 11136 . . . . . . . . . . . . . 14 2 ∈ ℕ
87 nnexpcl 12820 . . . . . . . . . . . . . 14 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
8886, 67, 87sylancr 694 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℕ)
8988nnred 10986 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℝ)
9089recnd 10019 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℂ)
9188nnne0d 11016 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ≠ 0)
9285, 90, 91divcan4d 10758 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)) = (𝐹𝑥))
9392eqcomd 2627 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) = (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)))
94 2cnd 11044 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ∈ ℂ)
95 2ne0 11064 . . . . . . . . . . 11 2 ≠ 0
9695a1i 11 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ≠ 0)
97 eluzelz 11648 . . . . . . . . . . 11 (𝑗 ∈ (ℤ𝑘) → 𝑗 ∈ ℤ)
9897adantl 482 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℤ)
9994, 96, 98exprecd 12963 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
10093, 99oveq12d 6628 . . . . . . . 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 10021 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℝ)
102101recnd 10019 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℂ)
103 1cnd 10007 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℂ)
104102, 103, 90, 91divsubdird 10791 . . . . . . . 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 2658 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)))
106 fllep1 12549 . . . . . . . . . 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 10006 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℝ)
109 reflcl 12544 . . . . . . . . . . 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 10575 . . . . . . . . 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 10299 . . . . . . . . . 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 11015 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 < (2↑𝑗))
116 lediv1 10839 . . . . . . . . 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 1327 . . . . . . . 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 4640 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
1201, 2, 3, 4mbfi1fseqlem2 23402 . . . . . . . . . 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 6155 . . . . . . . 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 6638 . . . . . . . . . . 11 (𝑗𝐽𝑥) ∈ V
124 vex 3192 . . . . . . . . . . 11 𝑗 ∈ V
125123, 124ifex 4133 . . . . . . . . . 10 if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) ∈ V
126 c0ex 9985 . . . . . . . . . 10 0 ∈ V
127125, 126ifex 4133 . . . . . . . . 9 if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) ∈ V
128 eqid 2621 . . . . . . . . . 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 6253 . . . . . . . . 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 693 . . . . . . . 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 2659 . . . . . . 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 761 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ∈ ℝ)
13315ad2antrr 761 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
13464nnred 10986 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℝ)
13511ad2antrr 761 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (𝐹𝑥))
138132, 66addge01d 10566 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℕ)
141140nnred 10986 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℝ)
142 simplrr 800 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)
143133, 141, 142ltled 10136 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑘)
144 eluzle 11651 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑘) → 𝑘𝑗)
145144adantl 482 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘𝑗)
146133, 141, 134, 143, 145letrd 10145 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑗)
147132, 133, 134, 139, 146letrd 10145 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ≤ 𝑗)
14882, 134absled 14110 . . . . . . . . . . 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 475 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗𝑥)
151149simprd 479 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥𝑗)
152134renegcld 10408 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗 ∈ ℝ)
153 elicc2 12187 . . . . . . . . . 10 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
154152, 134, 153syl2anc 692 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
15582, 150, 151, 154mpbir3and 1243 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ (-𝑗[,]𝑗))
156155iftrued 4071 . . . . . . 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 477 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑗𝑦 = 𝑥) → 𝑦 = 𝑥)
158157fveq2d 6157 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
159 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑗𝑦 = 𝑥) → 𝑚 = 𝑗)
160159oveq2d 6626 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (2↑𝑚) = (2↑𝑗))
161158, 160oveq12d 6628 . . . . . . . . . . . . . 14 ((𝑚 = 𝑗𝑦 = 𝑥) → ((𝐹𝑦) · (2↑𝑚)) = ((𝐹𝑥) · (2↑𝑗)))
162161fveq2d 6157 . . . . . . . . . . . . 13 ((𝑚 = 𝑗𝑦 = 𝑥) → (⌊‘((𝐹𝑦) · (2↑𝑚))) = (⌊‘((𝐹𝑥) · (2↑𝑗))))
163162, 160oveq12d 6628 . . . . . . . . . . . 12 ((𝑚 = 𝑗𝑦 = 𝑥) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
164 ovex 6638 . . . . . . . . . . . 12 ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ V
165163, 3, 164ovmpt2a 6751 . . . . . . . . . . 11 ((𝑗 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
16664, 82, 165syl2anc 692 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
167110, 88nndivred 11020 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ ℝ)
168 flle 12547 . . . . . . . . . . . . 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 10853 . . . . . . . . . . . . 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 1327 . . . . . . . . . . . 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 761 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℂ)
174173absge0d 14124 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (abs‘𝑥))
17566, 132addge02d 10567 . . . . . . . . . . . . 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 10145 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ≤ 𝑗)
178167, 66, 134, 172, 177letrd 10145 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ 𝑗)
179166, 178eqbrtrd 4640 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) ≤ 𝑗)
180179iftrued 4071 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = (𝑗𝐽𝑥))
181180, 166eqtrd 2655 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
182131, 156, 1813eqtrd 2659 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
183119, 65, 1823brtr4d 4650 . . . . 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 4640 . . . . 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 14312 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18617, 185rexlimddv 3029 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
187186ralrimiva 2961 . 2 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18836mptex 6446 . . . 4 (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ∈ V
1894, 188eqeltri 2694 . . 3 𝐺 ∈ V
190 feq1 5988 . . . 4 (𝑔 = 𝐺 → (𝑔:ℕ⟶dom ∫1𝐺:ℕ⟶dom ∫1))
191 fveq1 6152 . . . . . . 7 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
192191breq2d 4630 . . . . . 6 (𝑔 = 𝐺 → (0𝑝𝑟 ≤ (𝑔𝑛) ↔ 0𝑝𝑟 ≤ (𝐺𝑛)))
193 fveq1 6152 . . . . . . 7 (𝑔 = 𝐺 → (𝑔‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
194191, 193breq12d 4631 . . . . . 6 (𝑔 = 𝐺 → ((𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1)) ↔ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
195192, 194anbi12d 746 . . . . 5 (𝑔 = 𝐺 → ((0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
196195ralbidv 2981 . . . 4 (𝑔 = 𝐺 → (∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
197191fveq1d 6155 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛)‘𝑥) = ((𝐺𝑛)‘𝑥))
198197mpteq2dv 4710 . . . . . 6 (𝑔 = 𝐺 → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)))
199198breq1d 4628 . . . . 5 (𝑔 = 𝐺 → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
200199ralbidv 2981 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
201190, 196, 2003anbi123d 1396 . . 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 3289 . 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 1323 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  ifcif 4063   class class class wbr 4618  cmpt 4678  dom cdm 5079  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  𝑟 cofr 6856  cc 9885  cr 9886  0cc0 9887  1c1 9888   + caddc 9890   · cmul 9892  +∞cpnf 10022   < clt 10025  cle 10026  cmin 10217  -cneg 10218   / cdiv 10635  cn 10971  2c2 11021  0cn0 11243  cz 11328  cuz 11638  [,)cico 12126  [,]cicc 12127  cfl 12538  cexp 12807  abscabs 13915  cli 14156  MblFncmbf 23302  1citg1 23303  0𝑝c0p 23355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fi 8268  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-q 11740  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-ioo 12128  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-fl 12540  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-rlim 14161  df-sum 14358  df-rest 16011  df-topgen 16032  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-mopn 19670  df-top 20627  df-topon 20644  df-bases 20670  df-cmp 21109  df-ovol 23152  df-vol 23153  df-mbf 23307  df-itg1 23308  df-0p 23356
This theorem is referenced by:  mbfi1fseq  23407
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