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Theorem mbfi1fseqlem3 24312

Description: Lemma for mbfi1fseq 24316. (Contributed by Mario Carneiro, 16-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
mbfi1fseqlem3	⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))

Distinct variable groups: 𝑥,𝑚,𝑦,𝐹 𝑥,𝐺 𝑚,𝐽 𝜑,𝑚,𝑥,𝑦 𝐴,𝑚,𝑥,𝑦 Allowed substitution hints: 𝐺(𝑦,𝑚) 𝐽(𝑥,𝑦)

**Proof of Theorem mbfi1fseqlem3**
Step	Hyp	Ref	Expression
1		mbfi1fseq.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn)
2		mbfi1fseq.2	. . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
3		mbfi1fseq.3	. . . 4 ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))
4		mbfi1fseq.4	. . . 4 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
5	1, 2, 3, 4	mbfi1fseqlem2 24311	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
6	5	adantl 484	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
7		rge0ssre 12838	. . . . . . . . . . . . . . . . . 18 ⊢ (0[,)+∞) ⊆ ℝ
8		simpr 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
9		ffvelrn 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞))
10	2, 8, 9	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ (0[,)+∞))
11	7, 10	sseldi 3965	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ ℝ)
12		2nn 11704	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
13		nnnn0 11898	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
14		nnexpcl 13436	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
15	12, 13, 14	sylancr 589	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ)
16	15	ad2antrl 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ)
17	16	nnred 11647	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ)
18	11, 17	remulcld 10665	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ)
19		reflcl 13160	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ)
21	20, 16	nndivred 11685	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
22	21	ralrimivva 3191	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
23	3	fmpo 7760	. . . . . . . . . . . . 13 ⊢ (∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ↔ 𝐽:(ℕ × ℝ)⟶ℝ)
24	22, 23	sylib 220	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶ℝ)
25		fovrn 7312	. . . . . . . . . . . 12 ⊢ ((𝐽:(ℕ × ℝ)⟶ℝ ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
26	24, 25	syl3an1 1159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
27	26	3expa 1114	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
28		nnre 11639	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
29	28	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
30		nnnn0 11898	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
31		nnexpcl 13436	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
32	12, 30, 31	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑𝐴) ∈ ℕ)
33	32	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ)
34		nnre 11639	. . . . . . . . . . . 12 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℝ)
35		nngt0 11662	. . . . . . . . . . . 12 ⊢ ((2↑𝐴) ∈ ℕ → 0 < (2↑𝐴))
36	34, 35	jca 514	. . . . . . . . . . 11 ⊢ ((2↑𝐴) ∈ ℕ → ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴)))
37	33, 36	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴)))
38		lemul1 11486	. . . . . . . . . 10 ⊢ (((𝐴𝐽𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴))) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
39	27, 29, 37, 38	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
40	39	biimpa 479	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴)))
41		simpr 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
42	41	fveq2d 6669	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
43		simpl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑚 = 𝐴)
44	43	oveq2d 7166	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (2↑𝑚) = (2↑𝐴))
45	42, 44	oveq12d 7168	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((𝐹‘𝑦) · (2↑𝑚)) = ((𝐹‘𝑥) · (2↑𝐴)))
46	45	fveq2d 6669	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
47	46, 44	oveq12d 7168	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
48		ovex 7183	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ V
49	47, 3, 48	ovmpoa 7299	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
50	49	ad4ant23 751	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
51	50	oveq1d 7165	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)))
52	2	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞))
53	52	ffvelrnda 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
54		elrege0 12836	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
55	53, 54	sylib 220	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
56	55	simpld 497	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
57	33	nnred 11647	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℝ)
58	56, 57	remulcld 10665	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ)
59	33	nnnn0d 11949	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ₀)
60	59	nn0ge0d 11952	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (2↑𝐴))
61		mulge0 11152	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)) ∧ ((2↑𝐴) ∈ ℝ ∧ 0 ≤ (2↑𝐴))) → 0 ≤ ((𝐹‘𝑥) · (2↑𝐴)))
62	55, 57, 60, 61	syl12anc 834	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘𝑥) · (2↑𝐴)))
63		flge0nn0 13184	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑥) · (2↑𝐴))) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
64	58, 62, 63	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
65	64	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
66	65	nn0cnd 11951	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℂ)
67	33	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℕ)
68	67	nncnd 11648	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℂ)
69	67	nnne0d 11681	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ≠ 0)
70	66, 68, 69	divcan1d 11411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
71	51, 70	eqtrd 2856	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
72	71, 65	eqeltrd 2913	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ ℕ₀)
73		nn0uz 12274	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
74	72, 73	eleqtrdi 2923	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ_≥‘0))
75		nnmulcl 11655	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ (2↑𝐴) ∈ ℕ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
76	32, 75	mpdan 685	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐴 · (2↑𝐴)) ∈ ℕ)
77	76	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
78	77	adantr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℕ)
79	78	nnzd 12080	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℤ)
80		elfz5 12894	. . . . . . . . 9 ⊢ ((((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ_≥‘0) ∧ (𝐴 · (2↑𝐴)) ∈ ℤ) → (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
81	74, 79, 80	syl2anc 586	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
82	40, 81	mpbird 259	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
83		oveq1 7157	. . . . . . . 8 ⊢ (𝑚 = ((𝐴𝐽𝑥) · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
84		eqid 2821	. . . . . . . 8 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) = (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))
85		ovex 7183	. . . . . . . 8 ⊢ (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) ∈ V
86	83, 84, 85	fvmpt 6763	. . . . . . 7 ⊢ (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
87	82, 86	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
88	27	adantr 483	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℝ)
89	88	recnd 10663	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℂ)
90	89, 68, 69	divcan4d 11416	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) = (𝐴𝐽𝑥))
91	87, 90	eqtrd 2856	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (𝐴𝐽𝑥))
92		elfznn0 12994	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℕ₀)
93	92	nn0red 11950	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℝ)
94	32	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (2↑𝐴) ∈ ℕ)
95		nndivre 11672	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℝ ∧ (2↑𝐴) ∈ ℕ) → (𝑚 / (2↑𝐴)) ∈ ℝ)
96	93, 94, 95	syl2anr 598	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐴 · (2↑𝐴)))) → (𝑚 / (2↑𝐴)) ∈ ℝ)
97	96	fmpttd 6874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))):(0...(𝐴 · (2↑𝐴)))⟶ℝ)
98	97	ffnd 6510	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
99	98	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
100	99	adantr 483	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
101		fnfvelrn 6843	. . . . . 6 ⊢ (((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))) ∧ ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴)))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
102	100, 82, 101	syl2anc 586	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
103	91, 102	eqeltrrd 2914	. . . 4 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
104	77	nnnn0d 11949	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ₀)
105	104, 73	eleqtrdi 2923	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (ℤ_≥‘0))
106		eluzfz2 12909	. . . . . . . . 9 ⊢ ((𝐴 · (2↑𝐴)) ∈ (ℤ_≥‘0) → (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
107	105, 106	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
108		oveq1 7157	. . . . . . . . 9 ⊢ (𝑚 = (𝐴 · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
109		ovex 7183	. . . . . . . . 9 ⊢ ((𝐴 · (2↑𝐴)) / (2↑𝐴)) ∈ V
110	108, 84, 109	fvmpt 6763	. . . . . . . 8 ⊢ ((𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
111	107, 110	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
112	29	recnd 10663	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
113	33	nncnd 11648	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℂ)
114	33	nnne0d 11681	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ≠ 0)
115	112, 113, 114	divcan4d 11416	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 · (2↑𝐴)) / (2↑𝐴)) = 𝐴)
116	111, 115	eqtrd 2856	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = 𝐴)
117		fnfvelrn 6843	. . . . . . 7 ⊢ (((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))) ∧ (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴)))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
118	99, 107, 117	syl2anc 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
119	116, 118	eqeltrrd 2914	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
120	119	adantr 483	. . . 4 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
121	103, 120	ifclda 4501	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
122		eluzfz1 12908	. . . . . . 7 ⊢ ((𝐴 · (2↑𝐴)) ∈ (ℤ_≥‘0) → 0 ∈ (0...(𝐴 · (2↑𝐴))))
123	105, 122	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ (0...(𝐴 · (2↑𝐴))))
124		oveq1 7157	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 / (2↑𝐴)) = (0 / (2↑𝐴)))
125		ovex 7183	. . . . . . 7 ⊢ (0 / (2↑𝐴)) ∈ V
126	124, 84, 125	fvmpt 6763	. . . . . 6 ⊢ (0 ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = (0 / (2↑𝐴)))
127	123, 126	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = (0 / (2↑𝐴)))
128		nncn 11640	. . . . . . 7 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℂ)
129		nnne0 11665	. . . . . . 7 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ≠ 0)
130	128, 129	div0d 11409	. . . . . 6 ⊢ ((2↑𝐴) ∈ ℕ → (0 / (2↑𝐴)) = 0)
131	33, 130	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 / (2↑𝐴)) = 0)
132	127, 131	eqtrd 2856	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = 0)
133		fnfvelrn 6843	. . . . 5 ⊢ (((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))) ∧ 0 ∈ (0...(𝐴 · (2↑𝐴)))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
134	99, 123, 133	syl2anc 586	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
135	132, 134	eqeltrrd 2914	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
136	121, 135	ifcld 4512	. 2 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
137	6, 136	fmpt3d 6875	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ifcif 4467 class class class wbr 5059 ↦ cmpt 5139 × cxp 5548 ran crn 5551 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 ℝcr 10530 0cc0 10531 · cmul 10536 +∞cpnf 10666 < clt 10669 ≤ cle 10670 -cneg 10865 / cdiv 11291 ℕcn 11632 2c2 11686 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 [,)cico 12734 [,]cicc 12735 ...cfz 12886 ⌊cfl 13154 ↑cexp 13423 MblFncmbf 24209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-ico 12738 df-fz 12887 df-fl 13156 df-seq 13364 df-exp 13424

This theorem is referenced by: mbfi1fseqlem4 24313

W3C validator