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Theorem mbfi1fseqlem3 23290

Description: Lemma for mbfi1fseq 23294. (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		rge0ssre 12151	. . . . . . . . . . . . . . . . . . 19 ⊢ (0[,)+∞) ⊆ ℝ
2		mbfi1fseq.2	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
3		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
4		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞))
5	2, 3, 4	syl2an 493	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ (0[,)+∞))
6	1, 5	sseldi 3566	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ ℝ)
7		2nn 11062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℕ
8		nnnn0 11176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
9		nnexpcl 12735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
10	7, 8, 9	sylancr 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ)
11	10	ad2antrl 760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ)
12	11	nnred 10912	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ)
13	6, 12	remulcld 9949	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ)
14		reflcl 12459	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ)
16	15, 11	nndivred 10946	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
17	16	ralrimivva 2954	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
18		mbfi1fseq.3	. . . . . . . . . . . . . . 15 ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))
19	18	fmpt2 7126	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ↔ 𝐽:(ℕ × ℝ)⟶ℝ)
20	17, 19	sylib 207	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶ℝ)
21		fovrn 6702	. . . . . . . . . . . . 13 ⊢ ((𝐽:(ℕ × ℝ)⟶ℝ ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
22	20, 21	syl3an1 1351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
23	22	3expa 1257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
24		nnre 10904	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
25	24	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
26		nnnn0 11176	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
27		nnexpcl 12735	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
28	7, 26, 27	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (2↑𝐴) ∈ ℕ)
29	28	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ)
30		nnre 10904	. . . . . . . . . . . . 13 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℝ)
31		nngt0 10926	. . . . . . . . . . . . 13 ⊢ ((2↑𝐴) ∈ ℕ → 0 < (2↑𝐴))
32	30, 31	jca 553	. . . . . . . . . . . 12 ⊢ ((2↑𝐴) ∈ ℕ → ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴)))
33	29, 32	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴)))
34		lemul1 10754	. . . . . . . . . . 11 ⊢ (((𝐴𝐽𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴))) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
35	23, 25, 33, 34	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
36	35	biimpa 500	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴)))
37		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℕ)
38	37	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ℕ)
39		simplr 788	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → 𝑥 ∈ ℝ)
40		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
41	40	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
42		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑚 = 𝐴)
43	42	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (2↑𝑚) = (2↑𝐴))
44	41, 43	oveq12d 6567	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((𝐹‘𝑦) · (2↑𝑚)) = ((𝐹‘𝑥) · (2↑𝐴)))
45	44	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
46	45, 43	oveq12d 6567	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
47		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ V
48	46, 18, 47	ovmpt2a 6689	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
49	38, 39, 48	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
50	49	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)))
51	2	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞))
52	51	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
53		elrege0 12149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
54	52, 53	sylib 207	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
55	54	simpld 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
56	29	nnred 10912	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℝ)
57	55, 56	remulcld 9949	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ)
58	29	nnnn0d 11228	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ₀)
59	58	nn0ge0d 11231	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (2↑𝐴))
60		mulge0 10425	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)) ∧ ((2↑𝐴) ∈ ℝ ∧ 0 ≤ (2↑𝐴))) → 0 ≤ ((𝐹‘𝑥) · (2↑𝐴)))
61	54, 56, 59, 60	syl12anc 1316	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘𝑥) · (2↑𝐴)))
62		flge0nn0 12483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑥) · (2↑𝐴))) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
63	57, 61, 62	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
64	63	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
65	64	nn0cnd 11230	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℂ)
66	29	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℕ)
67	66	nncnd 10913	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℂ)
68	66	nnne0d 10942	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ≠ 0)
69	65, 67, 68	divcan1d 10681	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
70	50, 69	eqtrd 2644	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
71	70, 64	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ ℕ₀)
72		nn0uz 11598	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
73	71, 72	syl6eleq 2698	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ_≥‘0))
74		nnmulcl 10920	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ (2↑𝐴) ∈ ℕ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
75	28, 74	mpdan 699	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (𝐴 · (2↑𝐴)) ∈ ℕ)
76	75	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
77	76	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℕ)
78	77	nnzd 11357	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℤ)
79		elfz5 12205	. . . . . . . . . 10 ⊢ ((((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ_≥‘0) ∧ (𝐴 · (2↑𝐴)) ∈ ℤ) → (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
80	73, 78, 79	syl2anc 691	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
81	36, 80	mpbird 246	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
82		oveq1 6556	. . . . . . . . 9 ⊢ (𝑚 = ((𝐴𝐽𝑥) · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
83		eqid 2610	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) = (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))
84		ovex 6577	. . . . . . . . 9 ⊢ (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) ∈ V
85	82, 83, 84	fvmpt 6191	. . . . . . . 8 ⊢ (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
86	81, 85	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
87	23	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℝ)
88	87	recnd 9947	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℂ)
89	88, 67, 68	divcan4d 10686	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) = (𝐴𝐽𝑥))
90	86, 89	eqtrd 2644	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (𝐴𝐽𝑥))
91		elfznn0 12302	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℕ₀)
92	91	nn0red 11229	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℝ)
93	28	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (2↑𝐴) ∈ ℕ)
94		nndivre 10933	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℝ ∧ (2↑𝐴) ∈ ℕ) → (𝑚 / (2↑𝐴)) ∈ ℝ)
95	92, 93, 94	syl2anr 494	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐴 · (2↑𝐴)))) → (𝑚 / (2↑𝐴)) ∈ ℝ)
96	95, 83	fmptd 6292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))):(0...(𝐴 · (2↑𝐴)))⟶ℝ)
97		ffn 5958	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))):(0...(𝐴 · (2↑𝐴)))⟶ℝ → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
98	96, 97	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
99	98	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
100	99	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
101		fnfvelrn 6264	. . . . . . 7 ⊢ (((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, 81, 101	syl2anc 691	. . . . . 6 ⊢ ((((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	90, 102	eqeltrrd 2689	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
104	76	nnnn0d 11228	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ₀)
105	104, 72	syl6eleq 2698	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (ℤ_≥‘0))
106		eluzfz2 12220	. . . . . . . . . 10 ⊢ ((𝐴 · (2↑𝐴)) ∈ (ℤ_≥‘0) → (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
107	105, 106	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
108		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑚 = (𝐴 · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
109		ovex 6577	. . . . . . . . . 10 ⊢ ((𝐴 · (2↑𝐴)) / (2↑𝐴)) ∈ V
110	108, 83, 109	fvmpt 6191	. . . . . . . . 9 ⊢ ((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	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
112	25	recnd 9947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
113	29	nncnd 10913	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℂ)
114	29	nnne0d 10942	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ≠ 0)
115	112, 113, 114	divcan4d 10686	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 · (2↑𝐴)) / (2↑𝐴)) = 𝐴)
116	111, 115	eqtrd 2644	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = 𝐴)
117		fnfvelrn 6264	. . . . . . . 8 ⊢ (((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 691	. . . . . . 7 ⊢ (((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 2689	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
120	119	adantr 480	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
121	103, 120	ifclda 4070	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
122		eluzfz1 12219	. . . . . . . 8 ⊢ ((𝐴 · (2↑𝐴)) ∈ (ℤ_≥‘0) → 0 ∈ (0...(𝐴 · (2↑𝐴))))
123	105, 122	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ (0...(𝐴 · (2↑𝐴))))
124		oveq1 6556	. . . . . . . 8 ⊢ (𝑚 = 0 → (𝑚 / (2↑𝐴)) = (0 / (2↑𝐴)))
125		ovex 6577	. . . . . . . 8 ⊢ (0 / (2↑𝐴)) ∈ V
126	124, 83, 125	fvmpt 6191	. . . . . . 7 ⊢ (0 ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = (0 / (2↑𝐴)))
127	123, 126	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = (0 / (2↑𝐴)))
128		nncn 10905	. . . . . . . 8 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℂ)
129		nnne0 10930	. . . . . . . 8 ⊢ ((2↑𝐴) ∈ ℕ → (2↑𝐴) ≠ 0)
130	128, 129	div0d 10679	. . . . . . 7 ⊢ ((2↑𝐴) ∈ ℕ → (0 / (2↑𝐴)) = 0)
131	29, 130	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 / (2↑𝐴)) = 0)
132	127, 131	eqtrd 2644	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = 0)
133		fnfvelrn 6264	. . . . . 6 ⊢ (((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 691	. . . . 5 ⊢ (((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 2689	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
136	121, 135	ifcld 4081	. . 3 ⊢ (((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		eqid 2610	. . 3 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
138	136, 137	fmptd 6292	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
139		mbfi1fseq.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ MblFn)
140		mbfi1fseq.4	. . . . 5 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
141	139, 2, 18, 140	mbfi1fseqlem2 23289	. . . 4 ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
142	141	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
143	142	feq1d 5943	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((𝐺‘𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) ↔ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))))
144	138, 143	mpbird 246	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ran crn 5039 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 0cc0 9815 · cmul 9820 +∞cpnf 9950 < clt 9953 ≤ cle 9954 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 [,)cico 12048 [,]cicc 12049 ...cfz 12197 ⌊cfl 12453 ↑cexp 12722 MblFncmbf 23189

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-ico 12052 df-fz 12198 df-fl 12455 df-seq 12664 df-exp 12723

This theorem is referenced by: mbfi1fseqlem4 23291

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