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Theorem mbfi1fseqlem6 23781
Description: Lemma for mbfi1fseq 23782. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existential 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 23779 . 2 (𝜑𝐺:ℕ⟶dom ∫1)
61, 2, 3, 4mbfi1fseqlem5 23780 . . 3 ((𝜑𝑛 ∈ ℕ) → (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
76ralrimiva 3113 . 2 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
8 simpr 477 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
98recnd 10324 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
109abscld 14463 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (abs‘𝑥) ∈ ℝ)
112ffvelrnda 6551 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
12 elrege0 12485 . . . . . . . 8 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1311, 12sylib 209 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1413simpld 488 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
1510, 14readdcld 10325 . . . . 5 ((𝜑𝑥 ∈ ℝ) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
16 arch 11537 . . . . 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 2765 . . . . 5 (ℤ𝑘) = (ℤ𝑘)
19 nnz 11649 . . . . . 6 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
2019ad2antrl 719 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℤ)
21 nnuz 11926 . . . . . . . 8 ℕ = (ℤ‘1)
22 1zzd 11658 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 1 ∈ ℤ)
23 halfcn 11495 . . . . . . . . . 10 (1 / 2) ∈ ℂ
2423a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (1 / 2) ∈ ℂ)
25 halfre 11494 . . . . . . . . . . . 12 (1 / 2) ∈ ℝ
26 0re 10297 . . . . . . . . . . . . 13 0 ∈ ℝ
27 halfgt0 11496 . . . . . . . . . . . . 13 0 < (1 / 2)
2826, 25, 27ltleii 10416 . . . . . . . . . . . 12 0 ≤ (1 / 2)
29 absid 14324 . . . . . . . . . . . 12 (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2))
3025, 28, 29mp2an 683 . . . . . . . . . . 11 (abs‘(1 / 2)) = (1 / 2)
31 halflt1 11498 . . . . . . . . . . 11 (1 / 2) < 1
3230, 31eqbrtri 4832 . . . . . . . . . 10 (abs‘(1 / 2)) < 1
3332a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (abs‘(1 / 2)) < 1)
3424, 33expcnv 14883 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) ⇝ 0)
3514recnd 10324 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℂ)
36 nnex 11283 . . . . . . . . . 10 ℕ ∈ V
3736mptex 6681 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V
3837a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V)
39 nnnn0 11548 . . . . . . . . . . 11 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
4039adantl 473 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
41 oveq2 6852 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((1 / 2)↑𝑛) = ((1 / 2)↑𝑗))
42 eqid 2765 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))
43 ovex 6876 . . . . . . . . . . 11 ((1 / 2)↑𝑗) ∈ V
4441, 42, 43fvmpt 6473 . . . . . . . . . 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 13088 . . . . . . . . . 10 (((1 / 2) ∈ ℂ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℂ)
4723, 40, 46sylancr 581 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((1 / 2)↑𝑗) ∈ ℂ)
4845, 47eqeltrd 2844 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) ∈ ℂ)
4941oveq2d 6860 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝐹𝑥) − ((1 / 2)↑𝑛)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
50 eqid 2765 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))
51 ovex 6876 . . . . . . . . . . 11 ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ V
5249, 50, 51fvmpt 6473 . . . . . . . . . 10 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5352adantl 473 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5445oveq2d 6860 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5553, 54eqtr4d 2802 . . . . . . . 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 14657 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ ((𝐹𝑥) − 0))
5735subid1d 10637 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) − 0) = (𝐹𝑥))
5856, 57breqtrd 4837 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
5958adantr 472 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
6036mptex 6681 . . . . . 6 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V
6160a1i 11 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V)
62 simprl 787 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℕ)
63 eluznn 11962 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ)
6462, 63sylan 575 . . . . . . 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 717 . . . . . . 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 13087 . . . . . . . 8 (((1 / 2) ∈ ℝ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℝ)
6925, 67, 68sylancr 581 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) ∈ ℝ)
7066, 69resubcld 10714 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ ℝ)
7165, 70eqeltrd 2844 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ∈ ℝ)
72 fveq2 6377 . . . . . . . . 9 (𝑛 = 𝑗 → (𝐺𝑛) = (𝐺𝑗))
7372fveq1d 6379 . . . . . . . 8 (𝑛 = 𝑗 → ((𝐺𝑛)‘𝑥) = ((𝐺𝑗)‘𝑥))
74 eqid 2765 . . . . . . . 8 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))
75 fvex 6390 . . . . . . . 8 ((𝐺𝑗)‘𝑥) ∈ V
7673, 74, 75fvmpt 6473 . . . . . . 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 721 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐺:ℕ⟶dom ∫1)
7978, 64ffvelrnd 6552 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐺𝑗) ∈ dom ∫1)
80 i1ff 23737 . . . . . . . 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 717 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
8381, 82ffvelrnd 6552 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐺𝑗)‘𝑥) ∈ ℝ)
8477, 83eqeltrd 2844 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) ∈ ℝ)
8535ad2antrr 717 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ ℂ)
86 2nn 11347 . . . . . . . . . . . . . 14 2 ∈ ℕ
87 nnexpcl 13083 . . . . . . . . . . . . . 14 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
8886, 67, 87sylancr 581 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℕ)
8988nnred 11293 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℝ)
9089recnd 10324 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℂ)
9188nnne0d 11324 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ≠ 0)
9285, 90, 91divcan4d 11063 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)) = (𝐹𝑥))
9392eqcomd 2771 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) = (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)))
94 2cnd 11352 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ∈ ℂ)
95 2ne0 11385 . . . . . . . . . . 11 2 ≠ 0
9695a1i 11 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ≠ 0)
97 eluzelz 11899 . . . . . . . . . . 11 (𝑗 ∈ (ℤ𝑘) → 𝑗 ∈ ℤ)
9897adantl 473 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℤ)
9994, 96, 98exprecd 13226 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
10093, 99oveq12d 6862 . . . . . . . 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 10326 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℝ)
102101recnd 10324 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℂ)
103 1cnd 10290 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℂ)
104102, 103, 90, 91divsubdird 11096 . . . . . . . 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 2802 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)))
106 fllep1 12813 . . . . . . . . . 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 10296 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℝ)
109 reflcl 12808 . . . . . . . . . . 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 10880 . . . . . . . . 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 248 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹𝑥) · (2↑𝑗))))
113 peano2rem 10604 . . . . . . . . . 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 11323 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 < (2↑𝑗))
116 lediv1 11144 . . . . . . . . 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 1493 . . . . . . . 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 223 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
119105, 118eqbrtrd 4833 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
1201, 2, 3, 4mbfi1fseqlem2 23777 . . . . . . . . . 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 6379 . . . . . . . 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 6876 . . . . . . . . . . 11 (𝑗𝐽𝑥) ∈ V
124 vex 3353 . . . . . . . . . . 11 𝑗 ∈ V
125123, 124ifex 4293 . . . . . . . . . 10 if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) ∈ V
126 c0ex 10289 . . . . . . . . . 10 0 ∈ V
127125, 126ifex 4293 . . . . . . . . 9 if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) ∈ V
128 eqid 2765 . . . . . . . . . 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 6482 . . . . . . . . 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 580 . . . . . . . 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 2803 . . . . . . 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 717 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ∈ ℝ)
13315ad2antrr 717 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
13464nnred 11293 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℝ)
13511ad2antrr 717 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ (0[,)+∞))
136135, 12sylib 209 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
137136simprd 489 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (𝐹𝑥))
138132, 66addge01d 10871 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (0 ≤ (𝐹𝑥) ↔ (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥))))
139137, 138mpbid 223 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥)))
14062adantr 472 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℕ)
141140nnred 11293 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℝ)
142 simplrr 796 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)
143133, 141, 142ltled 10441 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑘)
144 eluzle 11902 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑘) → 𝑘𝑗)
145144adantl 473 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘𝑗)
146133, 141, 134, 143, 145letrd 10450 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑗)
147132, 133, 134, 139, 146letrd 10450 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ≤ 𝑗)
14882, 134absled 14457 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) ≤ 𝑗 ↔ (-𝑗𝑥𝑥𝑗)))
149147, 148mpbid 223 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (-𝑗𝑥𝑥𝑗))
150149simpld 488 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗𝑥)
151149simprd 489 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥𝑗)
152134renegcld 10713 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗 ∈ ℝ)
153 elicc2 12443 . . . . . . . . . 10 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
154152, 134, 153syl2anc 579 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
15582, 150, 151, 154mpbir3and 1442 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ (-𝑗[,]𝑗))
156155iftrued 4253 . . . . . . 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 6381 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
159 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑗𝑦 = 𝑥) → 𝑚 = 𝑗)
160159oveq2d 6860 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (2↑𝑚) = (2↑𝑗))
161158, 160oveq12d 6862 . . . . . . . . . . . . . 14 ((𝑚 = 𝑗𝑦 = 𝑥) → ((𝐹𝑦) · (2↑𝑚)) = ((𝐹𝑥) · (2↑𝑗)))
162161fveq2d 6381 . . . . . . . . . . . . 13 ((𝑚 = 𝑗𝑦 = 𝑥) → (⌊‘((𝐹𝑦) · (2↑𝑚))) = (⌊‘((𝐹𝑥) · (2↑𝑗))))
163162, 160oveq12d 6862 . . . . . . . . . . . 12 ((𝑚 = 𝑗𝑦 = 𝑥) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
164 ovex 6876 . . . . . . . . . . . 12 ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ V
165163, 3, 164ovmpt2a 6991 . . . . . . . . . . 11 ((𝑗 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
16664, 82, 165syl2anc 579 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
167110, 88nndivred 11328 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ ℝ)
168 flle 12811 . . . . . . . . . . . . 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 11158 . . . . . . . . . . . . 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 1493 . . . . . . . . . . . 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 248 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹𝑥))
1739ad2antrr 717 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℂ)
174173absge0d 14471 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (abs‘𝑥))
17566, 132addge02d 10872 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (0 ≤ (abs‘𝑥) ↔ (𝐹𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥))))
176174, 175mpbid 223 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥)))
17766, 133, 134, 176, 146letrd 10450 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ≤ 𝑗)
178167, 66, 134, 172, 177letrd 10450 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ 𝑗)
179166, 178eqbrtrd 4833 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) ≤ 𝑗)
180179iftrued 4253 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = (𝑗𝐽𝑥))
181180, 166eqtrd 2799 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
182131, 156, 1813eqtrd 2803 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
183119, 65, 1823brtr4d 4843 . . . . 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 4833 . . . . 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 14659 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18617, 185rexlimddv 3182 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
187186ralrimiva 3113 . 2 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18836mptex 6681 . . . 4 (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ∈ V
1894, 188eqeltri 2840 . . 3 𝐺 ∈ V
190 feq1 6206 . . . 4 (𝑔 = 𝐺 → (𝑔:ℕ⟶dom ∫1𝐺:ℕ⟶dom ∫1))
191 fveq1 6376 . . . . . . 7 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
192191breq2d 4823 . . . . . 6 (𝑔 = 𝐺 → (0𝑝𝑟 ≤ (𝑔𝑛) ↔ 0𝑝𝑟 ≤ (𝐺𝑛)))
193 fveq1 6376 . . . . . . 7 (𝑔 = 𝐺 → (𝑔‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
194191, 193breq12d 4824 . . . . . 6 (𝑔 = 𝐺 → ((𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1)) ↔ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
195192, 194anbi12d 624 . . . . 5 (𝑔 = 𝐺 → ((0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
196195ralbidv 3133 . . . 4 (𝑔 = 𝐺 → (∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
197191fveq1d 6379 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛)‘𝑥) = ((𝐺𝑛)‘𝑥))
198197mpteq2dv 4906 . . . . . 6 (𝑔 = 𝐺 → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)))
199198breq1d 4821 . . . . 5 (𝑔 = 𝐺 → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
200199ralbidv 3133 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
201190, 196, 2003anbi123d 1560 . . 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 3453 . 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 1490 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  ifcif 4245   class class class wbr 4811  cmpt 4890  dom cdm 5279  wf 6066  cfv 6070  (class class class)co 6844  cmpt2 6846  𝑟 cofr 7096  cc 10189  cr 10190  0cc0 10191  1c1 10192   + caddc 10194   · cmul 10196  +∞cpnf 10327   < clt 10330  cle 10331  cmin 10522  -cneg 10523   / cdiv 10940  cn 11276  2c2 11329  0cn0 11540  cz 11626  cuz 11889  [,)cico 12382  [,]cicc 12383  cfl 12802  cexp 13070  abscabs 14262  cli 14503  MblFncmbf 23675  1citg1 23676  0𝑝c0p 23730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097  df-ofr 7098  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-2o 7767  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-fi 8526  df-sup 8557  df-inf 8558  df-oi 8624  df-card 9018  df-cda 9245  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-q 11993  df-rp 12032  df-xneg 12149  df-xadd 12150  df-xmul 12151  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12537  df-fzo 12677  df-fl 12804  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-rlim 14508  df-sum 14705  df-rest 16352  df-topgen 16373  df-psmet 20014  df-xmet 20015  df-met 20016  df-bl 20017  df-mopn 20018  df-top 20981  df-topon 20998  df-bases 21033  df-cmp 21473  df-ovol 23525  df-vol 23526  df-mbf 23680  df-itg1 23681  df-0p 23731
This theorem is referenced by:  mbfi1fseq  23782
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