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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  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109   < clt 10112   ≤ cle 10113   − cmin 10304  -cneg 10305   / cdiv 10722  ℕcn 11058  2c2 11108  ℕ0cn0 11330  ℤcz 11415  ℤ≥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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