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Theorem mbfi1fseqlem3 24962

Description: Lemma for mbfi1fseq 24966. (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 24961	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
6	5	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
7		rge0ssre 13267	. . . . . . . . . . . . . . . . . 18 ⊢ (0[,)+∞) ⊆ ℝ
8		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
9		ffvelcdm 6998	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞))
10	2, 8, 9	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ (0[,)+∞))
11	7, 10	sselid 3928	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ ℝ)
12		2nn 12125	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
13		nnnn0 12319	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
14		nnexpcl 13874	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
15	12, 13, 14	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ)
16	15	ad2antrl 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ)
17	16	nnred 12067	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ)
18	11, 17	remulcld 11084	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ)
19		reflcl 13595	. . . . . . . . . . . . . . . 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 12106	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
22	21	ralrimivva 3193	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
23	3	fmpo 7954	. . . . . . . . . . . . 13 ⊢ (∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ↔ 𝐽:(ℕ × ℝ)⟶ℝ)
24	22, 23	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶ℝ)
25		fovcdm 7483	. . . . . . . . . . . 12 ⊢ ((𝐽:(ℕ × ℝ)⟶ℝ ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
26	24, 25	syl3an1 1162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
27	26	3expa 1117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
28		nnre 12059	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
29	28	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
30		nnnn0 12319	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
31		nnexpcl 13874	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
32	12, 30, 31	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑𝐴) ∈ ℕ)
33	32	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ)
34		nnre 12059	. . . . . . . . . . . 12 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℝ)
35		nngt0 12083	. . . . . . . . . . . 12 ⊢ ((2↑𝐴) ∈ ℕ → 0 < (2↑𝐴))
36	34, 35	jca 512	. . . . . . . . . . 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 11906	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
40	39	biimpa 477	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴)))
41		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
42	41	fveq2d 6815	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
43		simpl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑚 = 𝐴)
44	43	oveq2d 7332	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (2↑𝑚) = (2↑𝐴))
45	42, 44	oveq12d 7334	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((𝐹‘𝑦) · (2↑𝑚)) = ((𝐹‘𝑥) · (2↑𝐴)))
46	45	fveq2d 6815	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
47	46, 44	oveq12d 7334	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
48		ovex 7349	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ V
49	47, 3, 48	ovmpoa 7469	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
50	49	ad4ant23 750	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
51	50	oveq1d 7331	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)))
52	2	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞))
53	52	ffvelcdmda 7000	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
54		elrege0 13265	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
55	53, 54	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
56	55	simpld 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
57	33	nnred 12067	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℝ)
58	56, 57	remulcld 11084	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ)
59	33	nnnn0d 12372	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ₀)
60	59	nn0ge0d 12375	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (2↑𝐴))
61		mulge0 11572	. . . . . . . . . . . . . . . . 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 13619	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑥) · (2↑𝐴))) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
64	58, 62, 63	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
65	64	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
66	65	nn0cnd 12374	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℂ)
67	33	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℕ)
68	67	nncnd 12068	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℂ)
69	67	nnne0d 12102	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ≠ 0)
70	66, 68, 69	divcan1d 11831	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
71	51, 70	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
72	71, 65	eqeltrd 2837	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ ℕ₀)
73		nn0uz 12699	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
74	72, 73	eleqtrdi 2847	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ_≥‘0))
75		nnmulcl 12076	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ (2↑𝐴) ∈ ℕ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
76	32, 75	mpdan 684	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐴 · (2↑𝐴)) ∈ ℕ)
77	76	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
78	77	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℕ)
79	78	nnzd 12504	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℤ)
80		elfz5 13327	. . . . . . . . 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 584	. . . . . . . 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 256	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
83		oveq1 7323	. . . . . . . 8 ⊢ (𝑚 = ((𝐴𝐽𝑥) · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
84		eqid 2736	. . . . . . . 8 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) = (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))
85		ovex 7349	. . . . . . . 8 ⊢ (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) ∈ V
86	83, 84, 85	fvmpt 6914	. . . . . . 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 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℝ)
89	88	recnd 11082	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℂ)
90	89, 68, 69	divcan4d 11836	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) = (𝐴𝐽𝑥))
91	87, 90	eqtrd 2776	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (𝐴𝐽𝑥))
92		elfznn0 13428	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℕ₀)
93	92	nn0red 12373	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℝ)
94	32	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (2↑𝐴) ∈ ℕ)
95		nndivre 12093	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℝ ∧ (2↑𝐴) ∈ ℕ) → (𝑚 / (2↑𝐴)) ∈ ℝ)
96	93, 94, 95	syl2anr 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐴 · (2↑𝐴)))) → (𝑚 / (2↑𝐴)) ∈ ℝ)
97	96	fmpttd 7028	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))):(0...(𝐴 · (2↑𝐴)))⟶ℝ)
98	97	ffnd 6638	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
99	98	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
100	99	adantr 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
101		fnfvelrn 6997	. . . . . 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 584	. . . . 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 2838	. . . 4 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
104	77	nnnn0d 12372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ₀)
105	104, 73	eleqtrdi 2847	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (ℤ_≥‘0))
106		eluzfz2 13343	. . . . . . . . 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 7323	. . . . . . . . 9 ⊢ (𝑚 = (𝐴 · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
109		ovex 7349	. . . . . . . . 9 ⊢ ((𝐴 · (2↑𝐴)) / (2↑𝐴)) ∈ V
110	108, 84, 109	fvmpt 6914	. . . . . . . 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 11082	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
113	33	nncnd 12068	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℂ)
114	33	nnne0d 12102	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ≠ 0)
115	112, 113, 114	divcan4d 11836	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 · (2↑𝐴)) / (2↑𝐴)) = 𝐴)
116	111, 115	eqtrd 2776	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = 𝐴)
117		fnfvelrn 6997	. . . . . . 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 584	. . . . . 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 2838	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
120	119	adantr 481	. . . 4 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
121	103, 120	ifclda 4505	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
122		eluzfz1 13342	. . . . . . 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 7323	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 / (2↑𝐴)) = (0 / (2↑𝐴)))
125		ovex 7349	. . . . . . 7 ⊢ (0 / (2↑𝐴)) ∈ V
126	124, 84, 125	fvmpt 6914	. . . . . 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 12060	. . . . . . 7 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℂ)
129		nnne0 12086	. . . . . . 7 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ≠ 0)
130	128, 129	div0d 11829	. . . . . 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 2776	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = 0)
133		fnfvelrn 6997	. . . . 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 584	. . . 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 2838	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
136	121, 135	ifcld 4516	. 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 7029	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 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 ifcif 4470 class class class wbr 5086 ↦ cmpt 5169 × cxp 5605 ran crn 5608 Fn wfn 6460 ⟶wf 6461 ‘cfv 6465 (class class class)co 7316 ∈ cmpo 7318 ℝcr 10949 0cc0 10950 · cmul 10955 +∞cpnf 11085 < clt 11088 ≤ cle 11089 -cneg 11285 / cdiv 11711 ℕcn 12052 2c2 12107 ℕ₀cn0 12312 ℤcz 12398 ℤ_≥cuz 12661 [,)cico 13160 [,]cicc 13161 ...cfz 13318 ⌊cfl 13589 ↑cexp 13861 MblFncmbf 24858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-pre-sup 11028

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4850 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-om 7759 df-1st 7877 df-2nd 7878 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-er 8547 df-en 8783 df-dom 8784 df-sdom 8785 df-sup 9277 df-inf 9278 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-nn 12053 df-2 12115 df-n0 12313 df-z 12399 df-uz 12662 df-ico 13164 df-fz 13319 df-fl 13591 df-seq 13801 df-exp 13862

This theorem is referenced by: mbfi1fseqlem4 24963

W3C validator