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Theorem mbfi1fseqlem5 23292

Description: Lemma for mbfi1fseq 23294. Verify that 𝐺 describes an increasing sequence of positive functions. (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
mbfi1fseqlem5	⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (0_𝑝 ∘_𝑟 ≤ (𝐺‘𝐴) ∧ (𝐺‘𝐴) ∘_𝑟 ≤ (𝐺‘(𝐴 + 1))))

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 mbfi1fseqlem5**
Step	Hyp	Ref	Expression
1		mbfi1fseq.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
2	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞))
3	2	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
4		elrege0 12149	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
5	3, 4	sylib 207	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
6	5	simpld 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
7		2nn 11062	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
8		nnnn0 11176	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
9		nnexpcl 12735	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
10	7, 8, 9	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (2↑𝐴) ∈ ℕ)
11	10	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ)
12	11	nnred 10912	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℝ)
13	6, 12	remulcld 9949	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ)
14	11	nnnn0d 11228	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ₀)
15	14	nn0ge0d 11231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (2↑𝐴))
16		mulge0 10425	. . . . . . . . . . 11 ⊢ ((((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)) ∧ ((2↑𝐴) ∈ ℝ ∧ 0 ≤ (2↑𝐴))) → 0 ≤ ((𝐹‘𝑥) · (2↑𝐴)))
17	5, 12, 15, 16	syl12anc 1316	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘𝑥) · (2↑𝐴)))
18		flge0nn0 12483	. . . . . . . . . 10 ⊢ ((((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑥) · (2↑𝐴))) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
19	13, 17, 18	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℕ₀)
20	19	nn0red 11229	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℝ)
21	19	nn0ge0d 11231	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
22	11	nngt0d 10941	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 < (2↑𝐴))
23		divge0 10771	. . . . . . . 8 ⊢ ((((⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℝ ∧ 0 ≤ (⌊‘((𝐹‘𝑥) · (2↑𝐴)))) ∧ ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴))) → 0 ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
24	20, 21, 12, 22, 23	syl22anc 1319	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
25		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
26	25	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
27		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → 𝑚 = 𝐴)
28	27	oveq2d 6565	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (2↑𝑚) = (2↑𝐴))
29	26, 28	oveq12d 6567	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((𝐹‘𝑦) · (2↑𝑚)) = ((𝐹‘𝑥) · (2↑𝐴)))
30	29	fveq2d 6107	. . . . . . . . . 10 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
31	30, 28	oveq12d 6567	. . . . . . . . 9 ⊢ ((𝑚 = 𝐴 ∧ 𝑦 = 𝑥) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
32		mbfi1fseq.3	. . . . . . . . 9 ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))
33		ovex 6577	. . . . . . . . 9 ⊢ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ V
34	31, 32, 33	ovmpt2a 6689	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
35	34	adantll 746	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)))
36	24, 35	breqtrrd 4611	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐴𝐽𝑥))
37	8	nn0ge0d 11231	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 0 ≤ 𝐴)
38	37	ad2antlr 759	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ 𝐴)
39		breq2 4587	. . . . . . 7 ⊢ ((𝐴𝐽𝑥) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) → (0 ≤ (𝐴𝐽𝑥) ↔ 0 ≤ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴)))
40		breq2 4587	. . . . . . 7 ⊢ (𝐴 = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) → (0 ≤ 𝐴 ↔ 0 ≤ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴)))
41	39, 40	ifboth 4074	. . . . . 6 ⊢ ((0 ≤ (𝐴𝐽𝑥) ∧ 0 ≤ 𝐴) → 0 ≤ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴))
42	36, 38, 41	syl2anc 691	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴))
43		0le0 10987	. . . . 5 ⊢ 0 ≤ 0
44		breq2 4587	. . . . . 6 ⊢ (if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) → (0 ≤ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ↔ 0 ≤ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
45		breq2 4587	. . . . . 6 ⊢ (0 = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
46	44, 45	ifboth 4074	. . . . 5 ⊢ ((0 ≤ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∧ 0 ≤ 0) → 0 ≤ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
47	42, 43, 46	sylancl 693	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
48	47	ralrimiva 2949	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ∀𝑥 ∈ ℝ 0 ≤ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
49		0re 9919	. . . . . . 7 ⊢ 0 ∈ ℝ
50		fnconstg 6006	. . . . . . 7 ⊢ (0 ∈ ℝ → (ℂ × {0}) Fn ℂ)
51	49, 50	ax-mp 5	. . . . . 6 ⊢ (ℂ × {0}) Fn ℂ
52		df-0p 23243	. . . . . . 7 ⊢ 0_𝑝 = (ℂ × {0})
53	52	fneq1i 5899	. . . . . 6 ⊢ (0_𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
54	51, 53	mpbir 220	. . . . 5 ⊢ 0_𝑝 Fn ℂ
55	54	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 0_𝑝 Fn ℂ)
56		mbfi1fseq.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ MblFn)
57		mbfi1fseq.4	. . . . . . 7 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
58	56, 1, 32, 57	mbfi1fseqlem4 23291	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶dom ∫₁)
59	58	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴) ∈ dom ∫₁)
60		i1ff 23249	. . . . 5 ⊢ ((𝐺‘𝐴) ∈ dom ∫₁ → (𝐺‘𝐴):ℝ⟶ℝ)
61		ffn 5958	. . . . 5 ⊢ ((𝐺‘𝐴):ℝ⟶ℝ → (𝐺‘𝐴) Fn ℝ)
62	59, 60, 61	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴) Fn ℝ)
63		cnex 9896	. . . . 5 ⊢ ℂ ∈ V
64	63	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ℂ ∈ V)
65		reex 9906	. . . . 5 ⊢ ℝ ∈ V
66	65	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ℝ ∈ V)
67		ax-resscn 9872	. . . . 5 ⊢ ℝ ⊆ ℂ
68		sseqin2 3779	. . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℂ ∩ ℝ) = ℝ)
69	67, 68	mpbi 219	. . . 4 ⊢ (ℂ ∩ ℝ) = ℝ
70		0pval 23244	. . . . 5 ⊢ (𝑥 ∈ ℂ → (0_𝑝‘𝑥) = 0)
71	70	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (0_𝑝‘𝑥) = 0)
72	56, 1, 32, 57	mbfi1fseqlem2 23289	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
73	72	fveq1d 6105	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐺‘𝐴)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))‘𝑥))
74	73	ad2antlr 759	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝐴)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))‘𝑥))
75		simpr 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
76		rge0ssre 12151	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℝ
77		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
78		ffvelrn 6265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞))
79	1, 77, 78	syl2an 493	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ (0[,)+∞))
80	76, 79	sseldi 3566	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ ℝ)
81		nnnn0 11176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
82		nnexpcl 12735	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
83	7, 81, 82	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ)
84	83	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ)
85	84	nnred 10912	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ)
86	80, 85	remulcld 9949	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ)
87		reflcl 12459	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ)
88	86, 87	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ)
89	88, 84	nndivred 10946	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
90	89	ralrimivva 2954	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
91	32	fmpt2 7126	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ↔ 𝐽:(ℕ × ℝ)⟶ℝ)
92	90, 91	sylib 207	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶ℝ)
93		fovrn 6702	. . . . . . . . . 10 ⊢ ((𝐽:(ℕ × ℝ)⟶ℝ ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
94	92, 93	syl3an1 1351	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
95	94	3expa 1257	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
96		nnre 10904	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
97	96	ad2antlr 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
98	95, 97	ifcld 4081	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∈ ℝ)
99		ifcl 4080	. . . . . . 7 ⊢ ((if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ∈ ℝ)
100	98, 49, 99	sylancl 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ∈ ℝ)
101		eqid 2610	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
102	101	fvmpt2 6200	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))‘𝑥) = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
103	75, 100, 102	syl2anc 691	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))‘𝑥) = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
104	74, 103	eqtrd 2644	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝐴)‘𝑥) = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
105	55, 62, 64, 66, 69, 71, 104	ofrfval 6803	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (0_𝑝 ∘_𝑟 ≤ (𝐺‘𝐴) ↔ ∀𝑥 ∈ ℝ 0 ≤ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
106	48, 105	mpbird 246	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 0_𝑝 ∘_𝑟 ≤ (𝐺‘𝐴))
107	56, 1, 32	mbfi1fseqlem1 23288	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶(0[,)+∞))
108	107	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐽:(ℕ × ℝ)⟶(0[,)+∞))
109		peano2nn 10909	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
110	109	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 + 1) ∈ ℕ)
111	108, 110, 75	fovrnd 6704	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 + 1)𝐽𝑥) ∈ (0[,)+∞))
112		elrege0 12149	. . . . . . . . . . 11 ⊢ (((𝐴 + 1)𝐽𝑥) ∈ (0[,)+∞) ↔ (((𝐴 + 1)𝐽𝑥) ∈ ℝ ∧ 0 ≤ ((𝐴 + 1)𝐽𝑥)))
113	111, 112	sylib 207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐴 + 1)𝐽𝑥) ∈ ℝ ∧ 0 ≤ ((𝐴 + 1)𝐽𝑥)))
114	113	simpld 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 + 1)𝐽𝑥) ∈ ℝ)
115		min1 11894	. . . . . . . . . 10 ⊢ (((𝐴𝐽𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ (𝐴𝐽𝑥))
116	95, 97, 115	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ (𝐴𝐽𝑥))
117		2cn 10968	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
118	8	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℕ₀)
119		expp1 12729	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
120	117, 118, 119	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
121	120	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑(𝐴 + 1))) = (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · ((2↑𝐴) · 2)))
122	35, 95	eqeltrrd 2689	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ ℝ)
123	122	recnd 9947	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ ℂ)
124	12	recnd 9947	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℂ)
125		2cnd 10970	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
126	123, 124, 125	mulassd 9942	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) · 2) = (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · ((2↑𝐴) · 2)))
127	20	recnd 9947	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℂ)
128	11	nnne0d 10942	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ≠ 0)
129	127, 124, 128	divcan1d 10681	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) = (⌊‘((𝐹‘𝑥) · (2↑𝐴))))
130	129	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) · 2) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2))
131	121, 126, 130	3eqtr2d 2650	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑(𝐴 + 1))) = ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2))
132		flle 12462	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ≤ ((𝐹‘𝑥) · (2↑𝐴)))
133	13, 132	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ≤ ((𝐹‘𝑥) · (2↑𝐴)))
134		2re 10967	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
135		2pos 10989	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
136	134, 135	pm3.2i 470	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ ℝ ∧ 0 < 2)
137	136	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2 ∈ ℝ ∧ 0 < 2))
138		lemul1 10754	. . . . . . . . . . . . . . . 16 ⊢ (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℝ ∧ ((𝐹‘𝑥) · (2↑𝐴)) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) ≤ ((𝐹‘𝑥) · (2↑𝐴)) ↔ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ (((𝐹‘𝑥) · (2↑𝐴)) · 2)))
139	20, 13, 137, 138	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) ≤ ((𝐹‘𝑥) · (2↑𝐴)) ↔ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ (((𝐹‘𝑥) · (2↑𝐴)) · 2)))
140	133, 139	mpbid 221	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ (((𝐹‘𝑥) · (2↑𝐴)) · 2))
141	120	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑(𝐴 + 1))) = ((𝐹‘𝑥) · ((2↑𝐴) · 2)))
142	6	recnd 9947	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℂ)
143	142, 124, 125	mulassd 9942	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝐹‘𝑥) · (2↑𝐴)) · 2) = ((𝐹‘𝑥) · ((2↑𝐴) · 2)))
144	141, 143	eqtr4d 2647	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑(𝐴 + 1))) = (((𝐹‘𝑥) · (2↑𝐴)) · 2))
145	140, 144	breqtrrd 4611	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ ((𝐹‘𝑥) · (2↑(𝐴 + 1))))
146	110	nnnn0d 11228	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 + 1) ∈ ℕ₀)
147		nnexpcl 12735	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℕ ∧ (𝐴 + 1) ∈ ℕ₀) → (2↑(𝐴 + 1)) ∈ ℕ)
148	7, 146, 147	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑(𝐴 + 1)) ∈ ℕ)
149	148	nnred 10912	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑(𝐴 + 1)) ∈ ℝ)
150	6, 149	remulcld 9949	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑(𝐴 + 1))) ∈ ℝ)
151	13	flcld 12461	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℤ)
152		2z 11286	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
153		zmulcl 11303	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) ∈ ℤ ∧ 2 ∈ ℤ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ∈ ℤ)
154	151, 152, 153	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ∈ ℤ)
155		flge 12468	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑥) · (2↑(𝐴 + 1))) ∈ ℝ ∧ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ∈ ℤ) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ ((𝐹‘𝑥) · (2↑(𝐴 + 1))) ↔ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1))))))
156	150, 154, 155	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ ((𝐹‘𝑥) · (2↑(𝐴 + 1))) ↔ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1))))))
157	145, 156	mpbid 221	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) · 2) ≤ (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))))
158	131, 157	eqbrtrd 4605	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑(𝐴 + 1))) ≤ (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))))
159		reflcl 12459	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) · (2↑(𝐴 + 1))) ∈ ℝ → (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) ∈ ℝ)
160	150, 159	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) ∈ ℝ)
161	148	nngt0d 10941	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 < (2↑(𝐴 + 1)))
162		lemuldiv 10782	. . . . . . . . . . . 12 ⊢ ((((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ ℝ ∧ (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) ∈ ℝ ∧ ((2↑(𝐴 + 1)) ∈ ℝ ∧ 0 < (2↑(𝐴 + 1)))) → ((((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑(𝐴 + 1))) ≤ (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) ↔ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) / (2↑(𝐴 + 1)))))
163	122, 160, 149, 161, 162	syl112anc 1322	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑(𝐴 + 1))) ≤ (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) ↔ ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) / (2↑(𝐴 + 1)))))
164	158, 163	mpbid 221	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹‘𝑥) · (2↑𝐴))) / (2↑𝐴)) ≤ ((⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) / (2↑(𝐴 + 1))))
165		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = (𝐴 + 1) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
166	165	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = (𝐴 + 1) ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
167		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = (𝐴 + 1) ∧ 𝑦 = 𝑥) → 𝑚 = (𝐴 + 1))
168	167	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = (𝐴 + 1) ∧ 𝑦 = 𝑥) → (2↑𝑚) = (2↑(𝐴 + 1)))
169	166, 168	oveq12d 6567	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = (𝐴 + 1) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑦) · (2↑𝑚)) = ((𝐹‘𝑥) · (2↑(𝐴 + 1))))
170	169	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ ((𝑚 = (𝐴 + 1) ∧ 𝑦 = 𝑥) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) = (⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))))
171	170, 168	oveq12d 6567	. . . . . . . . . . . 12 ⊢ ((𝑚 = (𝐴 + 1) ∧ 𝑦 = 𝑥) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) / (2↑(𝐴 + 1))))
172		ovex 6577	. . . . . . . . . . . 12 ⊢ ((⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) / (2↑(𝐴 + 1))) ∈ V
173	171, 32, 172	ovmpt2a 6689	. . . . . . . . . . 11 ⊢ (((𝐴 + 1) ∈ ℕ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 1)𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) / (2↑(𝐴 + 1))))
174	110, 75, 173	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 + 1)𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑(𝐴 + 1)))) / (2↑(𝐴 + 1))))
175	164, 35, 174	3brtr4d 4615	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ≤ ((𝐴 + 1)𝐽𝑥))
176	98, 95, 114, 116, 175	letrd 10073	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ ((𝐴 + 1)𝐽𝑥))
177	110	nnred 10912	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 + 1) ∈ ℝ)
178		min2 11895	. . . . . . . . . 10 ⊢ (((𝐴𝐽𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ 𝐴)
179	95, 97, 178	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ 𝐴)
180	97	lep1d 10834	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ≤ (𝐴 + 1))
181	98, 97, 177, 179, 180	letrd 10073	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ (𝐴 + 1))
182		breq2 4587	. . . . . . . . 9 ⊢ (((𝐴 + 1)𝐽𝑥) = if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) → (if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ ((𝐴 + 1)𝐽𝑥) ↔ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1))))
183		breq2 4587	. . . . . . . . 9 ⊢ ((𝐴 + 1) = if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) → (if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ (𝐴 + 1) ↔ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1))))
184	182, 183	ifboth 4074	. . . . . . . 8 ⊢ ((if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ ((𝐴 + 1)𝐽𝑥) ∧ if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ (𝐴 + 1)) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)))
185	176, 181, 184	syl2anc 691	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)))
186	185	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (-𝐴[,]𝐴)) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)))
187		iftrue 4042	. . . . . . 7 ⊢ (𝑥 ∈ (-𝐴[,]𝐴) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴))
188	187	adantl 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (-𝐴[,]𝐴)) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴))
189	177	renegcld 10336	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → -(𝐴 + 1) ∈ ℝ)
190	97, 177	lenegd 10485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 ≤ (𝐴 + 1) ↔ -(𝐴 + 1) ≤ -𝐴))
191	180, 190	mpbid 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → -(𝐴 + 1) ≤ -𝐴)
192		iccss 12112	. . . . . . . . 9 ⊢ (((-(𝐴 + 1) ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) ∧ (-(𝐴 + 1) ≤ -𝐴 ∧ 𝐴 ≤ (𝐴 + 1))) → (-𝐴[,]𝐴) ⊆ (-(𝐴 + 1)[,](𝐴 + 1)))
193	189, 177, 191, 180, 192	syl22anc 1319	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (-𝐴[,]𝐴) ⊆ (-(𝐴 + 1)[,](𝐴 + 1)))
194	193	sselda 3568	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (-𝐴[,]𝐴)) → 𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)))
195	194	iftrued 4044	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (-𝐴[,]𝐴)) → if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0) = if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)))
196	186, 188, 195	3brtr4d 4615	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (-𝐴[,]𝐴)) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
197		iffalse 4045	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (-𝐴[,]𝐴) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) = 0)
198	197	adantl 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (-𝐴[,]𝐴)) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) = 0)
199	113	simprd 478	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐴 + 1)𝐽𝑥))
200	146	nn0ge0d 11231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐴 + 1))
201		breq2 4587	. . . . . . . . . 10 ⊢ (((𝐴 + 1)𝐽𝑥) = if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) → (0 ≤ ((𝐴 + 1)𝐽𝑥) ↔ 0 ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1))))
202		breq2 4587	. . . . . . . . . 10 ⊢ ((𝐴 + 1) = if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) → (0 ≤ (𝐴 + 1) ↔ 0 ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1))))
203	201, 202	ifboth 4074	. . . . . . . . 9 ⊢ ((0 ≤ ((𝐴 + 1)𝐽𝑥) ∧ 0 ≤ (𝐴 + 1)) → 0 ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)))
204	199, 200, 203	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)))
205		breq2 4587	. . . . . . . . 9 ⊢ (if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) = if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0) → (0 ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) ↔ 0 ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0)))
206		breq2 4587	. . . . . . . . 9 ⊢ (0 = if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0)))
207	205, 206	ifboth 4074	. . . . . . . 8 ⊢ ((0 ≤ if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) ∧ 0 ≤ 0) → 0 ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
208	204, 43, 207	sylancl 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
209	208	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (-𝐴[,]𝐴)) → 0 ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
210	198, 209	eqbrtrd 4605	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (-𝐴[,]𝐴)) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
211	196, 210	pm2.61dan 828	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
212	211	ralrimiva 2949	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
213		ffvelrn 6265	. . . . . 6 ⊢ ((𝐺:ℕ⟶dom ∫₁ ∧ (𝐴 + 1) ∈ ℕ) → (𝐺‘(𝐴 + 1)) ∈ dom ∫₁)
214	58, 109, 213	syl2an 493	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘(𝐴 + 1)) ∈ dom ∫₁)
215		i1ff 23249	. . . . 5 ⊢ ((𝐺‘(𝐴 + 1)) ∈ dom ∫₁ → (𝐺‘(𝐴 + 1)):ℝ⟶ℝ)
216		ffn 5958	. . . . 5 ⊢ ((𝐺‘(𝐴 + 1)):ℝ⟶ℝ → (𝐺‘(𝐴 + 1)) Fn ℝ)
217	214, 215, 216	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘(𝐴 + 1)) Fn ℝ)
218		inidm 3784	. . . 4 ⊢ (ℝ ∩ ℝ) = ℝ
219	56, 1, 32, 57	mbfi1fseqlem2 23289	. . . . . . 7 ⊢ ((𝐴 + 1) ∈ ℕ → (𝐺‘(𝐴 + 1)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0)))
220	219	fveq1d 6105	. . . . . 6 ⊢ ((𝐴 + 1) ∈ ℕ → ((𝐺‘(𝐴 + 1))‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))‘𝑥))
221	110, 220	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘(𝐴 + 1))‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))‘𝑥))
222	114, 177	ifcld 4081	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) ∈ ℝ)
223		ifcl 4080	. . . . . . 7 ⊢ ((if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0) ∈ ℝ)
224	222, 49, 223	sylancl 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0) ∈ ℝ)
225		eqid 2610	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
226	225	fvmpt2 6200	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0) ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))‘𝑥) = if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
227	75, 224, 226	syl2anc 691	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))‘𝑥) = if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
228	221, 227	eqtrd 2644	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘(𝐴 + 1))‘𝑥) = if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0))
229	62, 217, 66, 66, 218, 104, 228	ofrfval 6803	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((𝐺‘𝐴) ∘_𝑟 ≤ (𝐺‘(𝐴 + 1)) ↔ ∀𝑥 ∈ ℝ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ≤ if(𝑥 ∈ (-(𝐴 + 1)[,](𝐴 + 1)), if(((𝐴 + 1)𝐽𝑥) ≤ (𝐴 + 1), ((𝐴 + 1)𝐽𝑥), (𝐴 + 1)), 0)))
230	212, 229	mpbird 246	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴) ∘_𝑟 ≤ (𝐺‘(𝐴 + 1)))
231	106, 230	jca 553	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (0_𝑝 ∘_𝑟 ≤ (𝐺‘𝐴) ∧ (𝐺‘𝐴) ∘_𝑟 ≤ (𝐺‘(𝐴 + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ∘_𝑟 cofr 6794 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 +∞cpnf 9950 < clt 9953 ≤ cle 9954 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 [,)cico 12048 [,]cicc 12049 ⌊cfl 12453 ↑cexp 12722 MblFncmbf 23189 ∫₁citg1 23190 0_𝑝c0p 23242

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-inf2 8421 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-fal 1481 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-int 4411 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-se 4998 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 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-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 df-0p 23243

This theorem is referenced by: mbfi1fseqlem6 23293

W3C validator