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Theorem mbfi1fseqlem6 24313
Description: Lemma for mbfi1fseq 24314. 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 24311 . 2 (𝜑𝐺:ℕ⟶dom ∫1)
61, 2, 3, 4mbfi1fseqlem5 24312 . . 3 ((𝜑𝑛 ∈ ℕ) → (0𝑝r ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘r ≤ (𝐺‘(𝑛 + 1))))
76ralrimiva 3180 . 2 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘r ≤ (𝐺‘(𝑛 + 1))))
8 simpr 487 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
98recnd 10661 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
109abscld 14788 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (abs‘𝑥) ∈ ℝ)
112ffvelrnda 6844 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
12 elrege0 12834 . . . . . . . 8 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1311, 12sylib 220 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
1413simpld 497 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
1510, 14readdcld 10662 . . . . 5 ((𝜑𝑥 ∈ ℝ) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
16 arch 11886 . . . . 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 2819 . . . . 5 (ℤ𝑘) = (ℤ𝑘)
19 nnz 11996 . . . . . 6 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
2019ad2antrl 726 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℤ)
21 nnuz 12273 . . . . . . . 8 ℕ = (ℤ‘1)
22 1zzd 12005 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → 1 ∈ ℤ)
23 halfcn 11844 . . . . . . . . . 10 (1 / 2) ∈ ℂ
2423a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (1 / 2) ∈ ℂ)
25 halfre 11843 . . . . . . . . . . . 12 (1 / 2) ∈ ℝ
26 halfge0 11846 . . . . . . . . . . . 12 0 ≤ (1 / 2)
27 absid 14648 . . . . . . . . . . . 12 (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2))
2825, 26, 27mp2an 690 . . . . . . . . . . 11 (abs‘(1 / 2)) = (1 / 2)
29 halflt1 11847 . . . . . . . . . . 11 (1 / 2) < 1
3028, 29eqbrtri 5078 . . . . . . . . . 10 (abs‘(1 / 2)) < 1
3130a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (abs‘(1 / 2)) < 1)
3224, 31expcnv 15211 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) ⇝ 0)
3314recnd 10661 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℂ)
34 nnex 11636 . . . . . . . . . 10 ℕ ∈ V
3534mptex 6978 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V
3635a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ∈ V)
37 nnnn0 11896 . . . . . . . . . . 11 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
3837adantl 484 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
39 oveq2 7156 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((1 / 2)↑𝑛) = ((1 / 2)↑𝑗))
40 eqid 2819 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))
41 ovex 7181 . . . . . . . . . . 11 ((1 / 2)↑𝑗) ∈ V
4239, 40, 41fvmpt 6761 . . . . . . . . . 10 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) = ((1 / 2)↑𝑗))
4338, 42syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) = ((1 / 2)↑𝑗))
44 expcl 13439 . . . . . . . . . 10 (((1 / 2) ∈ ℂ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℂ)
4523, 38, 44sylancr 589 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((1 / 2)↑𝑗) ∈ ℂ)
4643, 45eqeltrd 2911 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) ∈ ℂ)
4739oveq2d 7164 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝐹𝑥) − ((1 / 2)↑𝑛)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
48 eqid 2819 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))
49 ovex 7181 . . . . . . . . . . 11 ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ V
5047, 48, 49fvmpt 6761 . . . . . . . . . 10 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5150adantl 484 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5243oveq2d 7164 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
5351, 52eqtr4d 2857 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)))
5421, 22, 32, 33, 36, 46, 53climsubc2 14987 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ ((𝐹𝑥) − 0))
5533subid1d 10978 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) − 0) = (𝐹𝑥))
5654, 55breqtrd 5083 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
5756adantr 483 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹𝑥))
5834mptex 6978 . . . . . 6 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V
5958a1i 11 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ∈ V)
60 simprl 769 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → 𝑘 ∈ ℕ)
61 eluznn 12310 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ)
6260, 61sylan 582 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ)
6362, 50syl 17 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹𝑥) − ((1 / 2)↑𝑗)))
6414ad2antrr 724 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ ℝ)
6562, 37syl 17 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℕ0)
66 reexpcl 13438 . . . . . . . 8 (((1 / 2) ∈ ℝ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℝ)
6725, 65, 66sylancr 589 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) ∈ ℝ)
6864, 67resubcld 11060 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ∈ ℝ)
6963, 68eqeltrd 2911 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ∈ ℝ)
70 fveq2 6663 . . . . . . . . 9 (𝑛 = 𝑗 → (𝐺𝑛) = (𝐺𝑗))
7170fveq1d 6665 . . . . . . . 8 (𝑛 = 𝑗 → ((𝐺𝑛)‘𝑥) = ((𝐺𝑗)‘𝑥))
72 eqid 2819 . . . . . . . 8 (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))
73 fvex 6676 . . . . . . . 8 ((𝐺𝑗)‘𝑥) ∈ V
7471, 72, 73fvmpt 6761 . . . . . . 7 (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((𝐺𝑗)‘𝑥))
7562, 74syl 17 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((𝐺𝑗)‘𝑥))
765ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐺:ℕ⟶dom ∫1)
7776, 62ffvelrnd 6845 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐺𝑗) ∈ dom ∫1)
78 i1ff 24269 . . . . . . . 8 ((𝐺𝑗) ∈ dom ∫1 → (𝐺𝑗):ℝ⟶ℝ)
7977, 78syl 17 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐺𝑗):ℝ⟶ℝ)
808ad2antrr 724 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
8179, 80ffvelrnd 6845 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐺𝑗)‘𝑥) ∈ ℝ)
8275, 81eqeltrd 2911 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) ∈ ℝ)
8333ad2antrr 724 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ ℂ)
84 2nn 11702 . . . . . . . . . . . . . 14 2 ∈ ℕ
85 nnexpcl 13434 . . . . . . . . . . . . . 14 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
8684, 65, 85sylancr 589 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℕ)
8786nnred 11645 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℝ)
8887recnd 10661 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ∈ ℂ)
8986nnne0d 11679 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (2↑𝑗) ≠ 0)
9083, 88, 89divcan4d 11414 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)) = (𝐹𝑥))
9190eqcomd 2825 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) = (((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)))
92 2cnd 11707 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ∈ ℂ)
93 2ne0 11733 . . . . . . . . . . 11 2 ≠ 0
9493a1i 11 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 2 ≠ 0)
95 eluzelz 12245 . . . . . . . . . . 11 (𝑗 ∈ (ℤ𝑘) → 𝑗 ∈ ℤ)
9695adantl 484 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℤ)
9792, 94, 96exprecd 13510 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
9891, 97oveq12d 7166 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹𝑥) · (2↑𝑗)) / (2↑𝑗)) − (1 / (2↑𝑗))))
9964, 87remulcld 10663 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℝ)
10099recnd 10661 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ∈ ℂ)
101 1cnd 10628 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℂ)
102100, 101, 88, 89divsubdird 11447 . . . . . . . 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))))
10398, 102eqtr4d 2857 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)))
104 fllep1 13163 . . . . . . . . . 10 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → ((𝐹𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) + 1))
10599, 104syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) + 1))
106 1red 10634 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 1 ∈ ℝ)
107 reflcl 13158 . . . . . . . . . . 11 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → (⌊‘((𝐹𝑥) · (2↑𝑗))) ∈ ℝ)
10899, 107syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (⌊‘((𝐹𝑥) · (2↑𝑗))) ∈ ℝ)
10999, 106, 108lesubaddd 11229 . . . . . . . . 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)))
110105, 109mpbird 259 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹𝑥) · (2↑𝑗))))
111 peano2rem 10945 . . . . . . . . . 10 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → (((𝐹𝑥) · (2↑𝑗)) − 1) ∈ ℝ)
11299, 111syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((𝐹𝑥) · (2↑𝑗)) − 1) ∈ ℝ)
11386nngt0d 11678 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 < (2↑𝑗))
114 lediv1 11497 . . . . . . . . 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))))
115112, 108, 87, 113, 114syl112anc 1369 . . . . . . . 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))))
116110, 115mpbid 234 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((((𝐹𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
117103, 116eqbrtrd 5079 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) − ((1 / 2)↑𝑗)) ≤ ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
1181, 2, 3, 4mbfi1fseqlem2 24309 . . . . . . . . . 10 (𝑗 ∈ ℕ → (𝐺𝑗) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0)))
11962, 118syl 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)))
120119fveq1d 6665 . . . . . . . 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))
121 ovex 7181 . . . . . . . . . . 11 (𝑗𝐽𝑥) ∈ V
122 vex 3496 . . . . . . . . . . 11 𝑗 ∈ V
123121, 122ifex 4513 . . . . . . . . . 10 if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) ∈ V
124 c0ex 10627 . . . . . . . . . 10 0 ∈ V
125123, 124ifex 4513 . . . . . . . . 9 if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) ∈ V
126 eqid 2819 . . . . . . . . . 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))
127126fvmpt2 6772 . . . . . . . . 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))
12880, 125, 127sylancl 588 . . . . . . . 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))
12975, 120, 1283eqtrd 2858 . . . . . . 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))
13010ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ∈ ℝ)
13115ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ∈ ℝ)
13262nnred 11645 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗 ∈ ℝ)
13311ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ∈ (0[,)+∞))
134133, 12sylib 220 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
135134simprd 498 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (𝐹𝑥))
136130, 64addge01d 11220 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (0 ≤ (𝐹𝑥) ↔ (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥))))
137135, 136mpbid 234 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥)))
13860adantr 483 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℕ)
139138nnred 11645 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘 ∈ ℝ)
140 simplrr 776 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)
141131, 139, 140ltled 10780 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑘)
142 eluzle 12248 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑘) → 𝑘𝑗)
143142adantl 484 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑘𝑗)
144131, 139, 132, 141, 143letrd 10789 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) + (𝐹𝑥)) ≤ 𝑗)
145130, 131, 132, 137, 144letrd 10789 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (abs‘𝑥) ≤ 𝑗)
14680, 132absled 14782 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((abs‘𝑥) ≤ 𝑗 ↔ (-𝑗𝑥𝑥𝑗)))
147145, 146mpbid 234 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (-𝑗𝑥𝑥𝑗))
148147simpld 497 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗𝑥)
149147simprd 498 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥𝑗)
150132renegcld 11059 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → -𝑗 ∈ ℝ)
151 elicc2 12793 . . . . . . . . . 10 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
152150, 132, 151syl2anc 586 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗𝑥𝑥𝑗)))
15380, 148, 149, 152mpbir3and 1337 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ (-𝑗[,]𝑗))
154153iftrued 4473 . . . . . . 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))
155 simpr 487 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑗𝑦 = 𝑥) → 𝑦 = 𝑥)
156155fveq2d 6667 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
157 simpl 485 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑗𝑦 = 𝑥) → 𝑚 = 𝑗)
158157oveq2d 7164 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑗𝑦 = 𝑥) → (2↑𝑚) = (2↑𝑗))
159156, 158oveq12d 7166 . . . . . . . . . . . . . 14 ((𝑚 = 𝑗𝑦 = 𝑥) → ((𝐹𝑦) · (2↑𝑚)) = ((𝐹𝑥) · (2↑𝑗)))
160159fveq2d 6667 . . . . . . . . . . . . 13 ((𝑚 = 𝑗𝑦 = 𝑥) → (⌊‘((𝐹𝑦) · (2↑𝑚))) = (⌊‘((𝐹𝑥) · (2↑𝑗))))
161160, 158oveq12d 7166 . . . . . . . . . . . 12 ((𝑚 = 𝑗𝑦 = 𝑥) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
162 ovex 7181 . . . . . . . . . . . 12 ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ V
163161, 3, 162ovmpoa 7297 . . . . . . . . . . 11 ((𝑗 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
16462, 80, 163syl2anc 586 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
165108, 86nndivred 11683 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ ℝ)
166 flle 13161 . . . . . . . . . . . . 13 (((𝐹𝑥) · (2↑𝑗)) ∈ ℝ → (⌊‘((𝐹𝑥) · (2↑𝑗))) ≤ ((𝐹𝑥) · (2↑𝑗)))
16799, 166syl 17 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (⌊‘((𝐹𝑥) · (2↑𝑗))) ≤ ((𝐹𝑥) · (2↑𝑗)))
168 ledivmul2 11511 . . . . . . . . . . . . 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))))
169108, 64, 87, 113, 168syl112anc 1369 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹𝑥) ↔ (⌊‘((𝐹𝑥) · (2↑𝑗))) ≤ ((𝐹𝑥) · (2↑𝑗))))
170167, 169mpbird 259 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹𝑥))
1719ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℂ)
172171absge0d 14796 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → 0 ≤ (abs‘𝑥))
17364, 130addge02d 11221 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (0 ≤ (abs‘𝑥) ↔ (𝐹𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥))))
174172, 173mpbid 234 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ≤ ((abs‘𝑥) + (𝐹𝑥)))
17564, 131, 132, 174, 144letrd 10789 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑥) ≤ 𝑗)
176165, 64, 132, 170, 175letrd 10789 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ 𝑗)
177164, 176eqbrtrd 5079 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑗𝐽𝑥) ≤ 𝑗)
178177iftrued 4473 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = (𝑗𝐽𝑥))
179178, 164eqtrd 2854 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
180129, 154, 1793eqtrd 2858 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) = ((⌊‘((𝐹𝑥) · (2↑𝑗))) / (2↑𝑗)))
181117, 63, 1803brtr4d 5089 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗))
182180, 170eqbrtrd 5079 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥))‘𝑗) ≤ (𝐹𝑥))
18318, 20, 57, 59, 69, 82, 181, 182climsqz 14989 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18417, 183rexlimddv 3289 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
185184ralrimiva 3180 . 2 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥))
18634mptex 6978 . . . 4 (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ∈ V
1874, 186eqeltri 2907 . . 3 𝐺 ∈ V
188 feq1 6488 . . . 4 (𝑔 = 𝐺 → (𝑔:ℕ⟶dom ∫1𝐺:ℕ⟶dom ∫1))
189 fveq1 6662 . . . . . . 7 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
190189breq2d 5069 . . . . . 6 (𝑔 = 𝐺 → (0𝑝r ≤ (𝑔𝑛) ↔ 0𝑝r ≤ (𝐺𝑛)))
191 fveq1 6662 . . . . . . 7 (𝑔 = 𝐺 → (𝑔‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
192189, 191breq12d 5070 . . . . . 6 (𝑔 = 𝐺 → ((𝑔𝑛) ∘r ≤ (𝑔‘(𝑛 + 1)) ↔ (𝐺𝑛) ∘r ≤ (𝐺‘(𝑛 + 1))))
193190, 192anbi12d 632 . . . . 5 (𝑔 = 𝐺 → ((0𝑝r ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ↔ (0𝑝r ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘r ≤ (𝐺‘(𝑛 + 1)))))
194193ralbidv 3195 . . . 4 (𝑔 = 𝐺 → (∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ↔ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝐺𝑛) ∧ (𝐺𝑛) ∘r ≤ (𝐺‘(𝑛 + 1)))))
195189fveq1d 6665 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛)‘𝑥) = ((𝐺𝑛)‘𝑥))
196195mpteq2dv 5153 . . . . . 6 (𝑔 = 𝐺 → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)))
197196breq1d 5067 . . . . 5 (𝑔 = 𝐺 → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
198197ralbidv 3195 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
199188, 194, 1983anbi123d 1430 . . 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))))
200187, 199spcev 3605 . 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)))
2015, 7, 185, 200syl3anc 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 208  wa 398  w3a 1082   = wceq 1531  wex 1774  wcel 2108  wne 3014  wral 3136  wrex 3137  Vcvv 3493  ifcif 4465   class class class wbr 5057  cmpt 5137  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7148  cmpo 7150  r cofr 7400  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  +∞cpnf 10664   < clt 10667  cle 10668  cmin 10862  -cneg 10863   / cdiv 11289  cn 11630  2c2 11684  0cn0 11889  cz 11973  cuz 12235  [,)cico 12732  [,]cicc 12733  cfl 13152  cexp 13421  abscabs 14585  cli 14833  MblFncmbf 24207  1citg1 24208  0𝑝c0p 24262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-ofr 7402  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-rest 16688  df-topgen 16709  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-top 21494  df-topon 21511  df-bases 21546  df-cmp 21987  df-ovol 24057  df-vol 24058  df-mbf 24212  df-itg1 24213  df-0p 24263
This theorem is referenced by:  mbfi1fseq  24314
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