Theorem mbfi1fseqlem6 23487

Description: Lemma for mbfi1fseq 23488. 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	⊢ (𝜑 → 𝐹 ∈ MblFn)
mbfi1fseq.2	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
mbfi1fseq.3	⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (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 (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))

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.

Step	Hyp	Ref	Expression
1		mbfi1fseq.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ MblFn)
2		mbfi1fseq.2	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
3		mbfi1fseq.3	. . 3 ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))
4		mbfi1fseq.4	. . 3 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
5	1, 2, 3, 4	mbfi1fseqlem4 23485	. 2 ⊢ (𝜑 → 𝐺:ℕ⟶dom ∫1)
6	1, 2, 3, 4	mbfi1fseqlem5 23486	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0𝑝 ∘𝑟 ≤ (𝐺‘𝑛) ∧ (𝐺‘𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
7	6	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝐺‘𝑛) ∧ (𝐺‘𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
8		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
9	8	recnd 10068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
10	9	abscld 14175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝑥) ∈ ℝ)
11	2	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
12		elrege0 12278	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
13	11, 12	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
14	13	simpld 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
15	10, 14	readdcld 10069	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs‘𝑥) + (𝐹‘𝑥)) ∈ ℝ)
16		arch 11289	. . . . 5 ⊢ (((abs‘𝑥) + (𝐹‘𝑥)) ∈ ℝ → ∃𝑘 ∈ ℕ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)
17	15, 16	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑘 ∈ ℕ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)
18		eqid 2622	. . . . 5 ⊢ (ℤ≥‘𝑘) = (ℤ≥‘𝑘)
19		nnz 11399	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
20	19	ad2antrl 764	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) → 𝑘 ∈ ℤ)
21		nnuz 11723	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
22		1zzd 11408	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ ℤ)
23		halfcn 11247	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℂ
24	23	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (1 / 2) ∈ ℂ)
25		halfre 11246	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
26		0re 10040	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
27		halfgt0 11248	. . . . . . . . . . . . 13 ⊢ 0 < (1 / 2)
28	26, 25, 27	ltleii 10160	. . . . . . . . . . . 12 ⊢ 0 ≤ (1 / 2)
29		absid 14036	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2))
30	25, 28, 29	mp2an 708	. . . . . . . . . . 11 ⊢ (abs‘(1 / 2)) = (1 / 2)
31		halflt1 11250	. . . . . . . . . . 11 ⊢ (1 / 2) < 1
32	30, 31	eqbrtri 4674	. . . . . . . . . 10 ⊢ (abs‘(1 / 2)) < 1
33	32	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘(1 / 2)) < 1)
34	24, 33	expcnv 14596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) ⇝ 0)
35	14	recnd 10068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℂ)
36		nnex 11026	. . . . . . . . . 10 ⊢ ℕ ∈ V
37	36	mptex 6486	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛))) ∈ V
38	37	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛))) ∈ V)
39		nnnn0 11299	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
40	39	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
41		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((1 / 2)↑𝑛) = ((1 / 2)↑𝑗))
42		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))
43		ovex 6678	. . . . . . . . . . 11 ⊢ ((1 / 2)↑𝑗) ∈ V
44	41, 42, 43	fvmpt 6282	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) = ((1 / 2)↑𝑗))
45	40, 44	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) = ((1 / 2)↑𝑗))
46		expcl 12878	. . . . . . . . . 10 ⊢ (((1 / 2) ∈ ℂ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℂ)
47	23, 40, 46	sylancr 695	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((1 / 2)↑𝑗) ∈ ℂ)
48	45, 47	eqeltrd 2701	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗) ∈ ℂ)
49	41	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝐹‘𝑥) − ((1 / 2)↑𝑛)) = ((𝐹‘𝑥) − ((1 / 2)↑𝑗)))
50		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛)))
51		ovex 6678	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) − ((1 / 2)↑𝑗)) ∈ V
52	49, 50, 51	fvmpt 6282	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹‘𝑥) − ((1 / 2)↑𝑗)))
53	52	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹‘𝑥) − ((1 / 2)↑𝑗)))
54	45	oveq2d 6666	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹‘𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)) = ((𝐹‘𝑥) − ((1 / 2)↑𝑗)))
55	53, 54	eqtr4d 2659	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹‘𝑥) − ((𝑛 ∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑗)))
56	21, 22, 34, 35, 38, 48, 55	climsubc2 14369	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛))) ⇝ ((𝐹‘𝑥) − 0))
57	35	subid1d 10381	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) − 0) = (𝐹‘𝑥))
58	56, 57	breqtrd 4679	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹‘𝑥))
59	58	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛))) ⇝ (𝐹‘𝑥))
60	36	mptex 6486	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ∈ V
61	60	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ∈ V)
62		simprl 794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) → 𝑘 ∈ ℕ)
63		eluznn 11758	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ ℕ)
64	62, 63	sylan 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ ℕ)
65	64, 52	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛)))‘𝑗) = ((𝐹‘𝑥) − ((1 / 2)↑𝑗)))
66	14	ad2antrr 762	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑥) ∈ ℝ)
67	64, 39	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ ℕ0)
68		reexpcl 12877	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ 𝑗 ∈ ℕ0) → ((1 / 2)↑𝑗) ∈ ℝ)
69	25, 67, 68	sylancr 695	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((1 / 2)↑𝑗) ∈ ℝ)
70	66, 69	resubcld 10458	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) − ((1 / 2)↑𝑗)) ∈ ℝ)
71	65, 70	eqeltrd 2701	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ∈ ℝ)
72		fveq2 6191	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝐺‘𝑛) = (𝐺‘𝑗))
73	72	fveq1d 6193	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ((𝐺‘𝑛)‘𝑥) = ((𝐺‘𝑗)‘𝑥))
74		eqid 2622	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))
75		fvex 6201	. . . . . . . 8 ⊢ ((𝐺‘𝑗)‘𝑥) ∈ V
76	73, 74, 75	fvmpt 6282	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))‘𝑗) = ((𝐺‘𝑗)‘𝑥))
77	64, 76	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))‘𝑗) = ((𝐺‘𝑗)‘𝑥))
78	5	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝐺:ℕ⟶dom ∫1)
79	78, 64	ffvelrnd 6360	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐺‘𝑗) ∈ dom ∫1)
80		i1ff 23443	. . . . . . . 8 ⊢ ((𝐺‘𝑗) ∈ dom ∫1 → (𝐺‘𝑗):ℝ⟶ℝ)
81	79, 80	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐺‘𝑗):ℝ⟶ℝ)
82	8	ad2antrr 762	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ ℝ)
83	81, 82	ffvelrnd 6360	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐺‘𝑗)‘𝑥) ∈ ℝ)
84	77, 83	eqeltrd 2701	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))‘𝑗) ∈ ℝ)
85	35	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑥) ∈ ℂ)
86		2nn 11185	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
87		nnexpcl 12873	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
88	86, 67, 87	sylancr 695	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (2↑𝑗) ∈ ℕ)
89	88	nnred 11035	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (2↑𝑗) ∈ ℝ)
90	89	recnd 10068	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (2↑𝑗) ∈ ℂ)
91	88	nnne0d 11065	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (2↑𝑗) ≠ 0)
92	85, 90, 91	divcan4d 10807	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((𝐹‘𝑥) · (2↑𝑗)) / (2↑𝑗)) = (𝐹‘𝑥))
93	92	eqcomd 2628	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑥) = (((𝐹‘𝑥) · (2↑𝑗)) / (2↑𝑗)))
94		2cnd 11093	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 2 ∈ ℂ)
95		2ne0 11113	. . . . . . . . . . 11 ⊢ 2 ≠ 0
96	95	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 2 ≠ 0)
97		eluzelz 11697	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑘) → 𝑗 ∈ ℤ)
98	97	adantl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ ℤ)
99	94, 96, 98	exprecd 13016	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
100	93, 99	oveq12d 6668	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹‘𝑥) · (2↑𝑗)) / (2↑𝑗)) − (1 / (2↑𝑗))))
101	66, 89	remulcld 10070	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) · (2↑𝑗)) ∈ ℝ)
102	101	recnd 10068	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) · (2↑𝑗)) ∈ ℂ)
103		1cnd 10056	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 1 ∈ ℂ)
104	102, 103, 90, 91	divsubdird 10840	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((((𝐹‘𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) = ((((𝐹‘𝑥) · (2↑𝑗)) / (2↑𝑗)) − (1 / (2↑𝑗))))
105	100, 104	eqtr4d 2659	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) − ((1 / 2)↑𝑗)) = ((((𝐹‘𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)))
106		fllep1 12602	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) · (2↑𝑗)) ∈ ℝ → ((𝐹‘𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) + 1))
107	101, 106	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) + 1))
108		1red 10055	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 1 ∈ ℝ)
109		reflcl 12597	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) · (2↑𝑗)) ∈ ℝ → (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ∈ ℝ)
110	101, 109	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ∈ ℝ)
111	101, 108, 110	lesubaddd 10624	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((((𝐹‘𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ↔ ((𝐹‘𝑥) · (2↑𝑗)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) + 1)))
112	107, 111	mpbird 247	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((𝐹‘𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑗))))
113		peano2rem 10348	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) · (2↑𝑗)) ∈ ℝ → (((𝐹‘𝑥) · (2↑𝑗)) − 1) ∈ ℝ)
114	101, 113	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((𝐹‘𝑥) · (2↑𝑗)) − 1) ∈ ℝ)
115	88	nngt0d 11064	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 0 < (2↑𝑗))
116		lediv1 10888	. . . . . . . . 9 ⊢ (((((𝐹‘𝑥) · (2↑𝑗)) − 1) ∈ ℝ ∧ (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ∈ ℝ ∧ ((2↑𝑗) ∈ ℝ ∧ 0 < (2↑𝑗))) → ((((𝐹‘𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ↔ ((((𝐹‘𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗))))
117	114, 110, 89, 115, 116	syl112anc 1330	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((((𝐹‘𝑥) · (2↑𝑗)) − 1) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ↔ ((((𝐹‘𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗))))
118	112, 117	mpbid 222	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((((𝐹‘𝑥) · (2↑𝑗)) − 1) / (2↑𝑗)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)))
119	105, 118	eqbrtrd 4675	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) − ((1 / 2)↑𝑗)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)))
120	1, 2, 3, 4	mbfi1fseqlem2 23483	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝐺‘𝑗) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0)))
121	64, 120	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐺‘𝑗) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0)))
122	121	fveq1d 6193	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐺‘𝑗)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))‘𝑥))
123		ovex 6678	. . . . . . . . . . 11 ⊢ (𝑗𝐽𝑥) ∈ V
124		vex 3203	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
125	123, 124	ifex 4156	. . . . . . . . . 10 ⊢ if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) ∈ V
126		c0ex 10034	. . . . . . . . . 10 ⊢ 0 ∈ V
127	125, 126	ifex 4156	. . . . . . . . 9 ⊢ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) ∈ V
128		eqid 2622	. . . . . . . . . 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))
129	128	fvmpt2 6291	. . . . . . . . 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))
130	82, 127, 129	sylancl 694	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))‘𝑥) = if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))
131	77, 122, 130	3eqtrd 2660	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))‘𝑗) = if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0))
132	10	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (abs‘𝑥) ∈ ℝ)
133	15	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((abs‘𝑥) + (𝐹‘𝑥)) ∈ ℝ)
134	64	nnred 11035	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ ℝ)
135	11	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑥) ∈ (0[,)+∞))
136	135, 12	sylib 208	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
137	136	simprd 479	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 0 ≤ (𝐹‘𝑥))
138	132, 66	addge01d 10615	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (0 ≤ (𝐹‘𝑥) ↔ (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹‘𝑥))))
139	137, 138	mpbid 222	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (abs‘𝑥) ≤ ((abs‘𝑥) + (𝐹‘𝑥)))
140	62	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑘 ∈ ℕ)
141	140	nnred 11035	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑘 ∈ ℝ)
142		simplrr 801	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)
143	133, 141, 142	ltled 10185	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((abs‘𝑥) + (𝐹‘𝑥)) ≤ 𝑘)
144		eluzle 11700	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ≥‘𝑘) → 𝑘 ≤ 𝑗)
145	144	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑘 ≤ 𝑗)
146	133, 141, 134, 143, 145	letrd 10194	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((abs‘𝑥) + (𝐹‘𝑥)) ≤ 𝑗)
147	132, 133, 134, 139, 146	letrd 10194	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (abs‘𝑥) ≤ 𝑗)
148	82, 134	absled 14169	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((abs‘𝑥) ≤ 𝑗 ↔ (-𝑗 ≤ 𝑥 ∧ 𝑥 ≤ 𝑗)))
149	147, 148	mpbid 222	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (-𝑗 ≤ 𝑥 ∧ 𝑥 ≤ 𝑗))
150	149	simpld 475	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → -𝑗 ≤ 𝑥)
151	149	simprd 479	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ≤ 𝑗)
152	134	renegcld 10457	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → -𝑗 ∈ ℝ)
153		elicc2 12238	. . . . . . . . . 10 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗 ≤ 𝑥 ∧ 𝑥 ≤ 𝑗)))
154	152, 134, 153	syl2anc 693	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑥 ∈ (-𝑗[,]𝑗) ↔ (𝑥 ∈ ℝ ∧ -𝑗 ≤ 𝑥 ∧ 𝑥 ≤ 𝑗)))
155	82, 150, 151, 154	mpbir3and 1245	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ (-𝑗[,]𝑗))
156	155	iftrued 4094	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → if(𝑥 ∈ (-𝑗[,]𝑗), if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗), 0) = if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗))
157		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝑗 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
158	157	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = 𝑗 ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
159		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝑗 ∧ 𝑦 = 𝑥) → 𝑚 = 𝑗)
160	159	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = 𝑗 ∧ 𝑦 = 𝑥) → (2↑𝑚) = (2↑𝑗))
161	158, 160	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑗 ∧ 𝑦 = 𝑥) → ((𝐹‘𝑦) · (2↑𝑚)) = ((𝐹‘𝑥) · (2↑𝑗)))
162	161	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑗 ∧ 𝑦 = 𝑥) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) = (⌊‘((𝐹‘𝑥) · (2↑𝑗))))
163	162, 160	oveq12d 6668	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑗 ∧ 𝑦 = 𝑥) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)))
164		ovex 6678	. . . . . . . . . . . 12 ⊢ ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ V
165	163, 3, 164	ovmpt2a 6791	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑗𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)))
166	64, 82, 165	syl2anc 693	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑗𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)))
167	110, 88	nndivred 11069	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)) ∈ ℝ)
168		flle 12600	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) · (2↑𝑗)) ∈ ℝ → (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ≤ ((𝐹‘𝑥) · (2↑𝑗)))
169	101, 168	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ≤ ((𝐹‘𝑥) · (2↑𝑗)))
170		ledivmul2 10902	. . . . . . . . . . . . 13 ⊢ (((⌊‘((𝐹‘𝑥) · (2↑𝑗))) ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ ∧ ((2↑𝑗) ∈ ℝ ∧ 0 < (2↑𝑗))) → (((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹‘𝑥) ↔ (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ≤ ((𝐹‘𝑥) · (2↑𝑗))))
171	110, 66, 89, 115, 170	syl112anc 1330	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹‘𝑥) ↔ (⌊‘((𝐹‘𝑥) · (2↑𝑗))) ≤ ((𝐹‘𝑥) · (2↑𝑗))))
172	169, 171	mpbird 247	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ (𝐹‘𝑥))
173	9	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ ℂ)
174	173	absge0d 14183	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 0 ≤ (abs‘𝑥))
175	66, 132	addge02d 10616	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (0 ≤ (abs‘𝑥) ↔ (𝐹‘𝑥) ≤ ((abs‘𝑥) + (𝐹‘𝑥))))
176	174, 175	mpbid 222	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑥) ≤ ((abs‘𝑥) + (𝐹‘𝑥)))
177	66, 133, 134, 176, 146	letrd 10194	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑥) ≤ 𝑗)
178	167, 66, 134, 172, 177	letrd 10194	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)) ≤ 𝑗)
179	166, 178	eqbrtrd 4675	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑗𝐽𝑥) ≤ 𝑗)
180	179	iftrued 4094	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = (𝑗𝐽𝑥))
181	180, 166	eqtrd 2656	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → if((𝑗𝐽𝑥) ≤ 𝑗, (𝑗𝐽𝑥), 𝑗) = ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)))
182	131, 156, 181	3eqtrd 2660	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))‘𝑗) = ((⌊‘((𝐹‘𝑥) · (2↑𝑗))) / (2↑𝑗)))
183	119, 65, 182	3brtr4d 4685	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑥) − ((1 / 2)↑𝑛)))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))‘𝑗))
184	182, 172	eqbrtrd 4675	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥))‘𝑗) ≤ (𝐹‘𝑥))
185	18, 20, 59, 61, 71, 84, 183, 184	climsqz 14371	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ ((abs‘𝑥) + (𝐹‘𝑥)) < 𝑘)) → (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
186	17, 185	rexlimddv 3035	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
187	186	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
188	36	mptex 6486	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ∈ V
189	4, 188	eqeltri 2697	. . 3 ⊢ 𝐺 ∈ V
190		feq1 6026	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔:ℕ⟶dom ∫1 ↔ 𝐺:ℕ⟶dom ∫1))
191		fveq1 6190	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑛) = (𝐺‘𝑛))
192	191	breq2d 4665	. . . . . 6 ⊢ (𝑔 = 𝐺 → (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ↔ 0𝑝 ∘𝑟 ≤ (𝐺‘𝑛)))
193		fveq1 6190	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
194	191, 193	breq12d 4666	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1)) ↔ (𝐺‘𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))))
195	192, 194	anbi12d 747	. . . . 5 ⊢ (𝑔 = 𝐺 → ((0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ (0𝑝 ∘𝑟 ≤ (𝐺‘𝑛) ∧ (𝐺‘𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
196	195	ralbidv 2986	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ↔ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝐺‘𝑛) ∧ (𝐺‘𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1)))))
197	191	fveq1d 6193	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑛)‘𝑥) = ((𝐺‘𝑛)‘𝑥))
198	197	mpteq2dv 4745	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)))
199	198	breq1d 4663	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
200	199	ralbidv 2986	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
201	190, 196, 200	3anbi123d 1399	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ↔ (𝐺:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝐺‘𝑛) ∧ (𝐺‘𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
202	189, 201	spcev 3300	. 2 ⊢ ((𝐺:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝐺‘𝑛) ∧ (𝐺‘𝑛) ∘𝑟 ≤ (𝐺‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
203	5, 7, 187, 202	syl3anc 1326	1 ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ∘_𝑟 cofr 6896  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  +∞cpnf 10071   < clt 10074   ≤ cle 10075   − cmin 10266  -cneg 10267   / cdiv 10684  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  [,)cico 12177  [,]cicc 12178  ⌊cfl 12591  ↑cexp 12860  abscabs 13974   ⇝ cli 14215  MblFncmbf 23383  ∫₁citg1 23384  0_𝑝c0p 23436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-0p 23437

This theorem is referenced by:  mbfi1fseq  23488