aaliou3lem2 - Metamath Proof Explorer

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Theorem aaliou3lem2 26403

Description: Lemma for aaliou3 26411. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
aaliou3lem.a	⊢ 𝐺 = (𝑐 ∈ (ℤ_≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))))
aaliou3lem.b	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))

**Assertion**
Ref	Expression
aaliou3lem2	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵)))

Distinct variable groups: 𝐹,𝑐 𝐴,𝑎,𝑐 𝐵,𝑎,𝑐 𝐺,𝑎 Allowed substitution hints: 𝐹(𝑎) 𝐺(𝑐)

Proof of Theorem aaliou3lem2 Dummy variables 𝑏 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluznn 12983	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 𝐵 ∈ ℕ)
2		fveq2 6920	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (!‘𝑎) = (!‘𝐵))
3	2	negeqd 11530	. . . . . . 7 ⊢ (𝑎 = 𝐵 → -(!‘𝑎) = -(!‘𝐵))
4	3	oveq2d 7464	. . . . . 6 ⊢ (𝑎 = 𝐵 → (2↑-(!‘𝑎)) = (2↑-(!‘𝐵)))
5		aaliou3lem.b	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
6		ovex 7481	. . . . . 6 ⊢ (2↑-(!‘𝐵)) ∈ V
7	4, 5, 6	fvmpt 7029	. . . . 5 ⊢ (𝐵 ∈ ℕ → (𝐹‘𝐵) = (2↑-(!‘𝐵)))
8	1, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) = (2↑-(!‘𝐵)))
9		2rp 13062	. . . . 5 ⊢ 2 ∈ ℝ⁺
10	1	nnnn0d 12613	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 𝐵 ∈ ℕ₀)
11	10	faccld 14333	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (!‘𝐵) ∈ ℕ)
12	11	nnzd 12666	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (!‘𝐵) ∈ ℤ)
13	12	znegcld 12749	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → -(!‘𝐵) ∈ ℤ)
14		rpexpcl 14131	. . . . 5 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐵) ∈ ℤ) → (2↑-(!‘𝐵)) ∈ ℝ⁺)
15	9, 13, 14	sylancr 586	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐵)) ∈ ℝ⁺)
16	8, 15	eqeltrd 2844	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ ℝ⁺)
17	16	rpred 13099	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ ℝ)
18	16	rpgt0d 13102	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 0 < (𝐹‘𝐵))
19		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝐹‘𝑏) = (𝐹‘𝐴))
20		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝐺‘𝑏) = (𝐺‘𝐴))
21	19, 20	breq12d 5179	. . . . 5 ⊢ (𝑏 = 𝐴 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝐴) ≤ (𝐺‘𝐴)))
22	21	imbi2d 340	. . . 4 ⊢ (𝑏 = 𝐴 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝐴) ≤ (𝐺‘𝐴))))
23		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
24		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝑑 → (𝐺‘𝑏) = (𝐺‘𝑑))
25	23, 24	breq12d 5179	. . . . 5 ⊢ (𝑏 = 𝑑 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝑑) ≤ (𝐺‘𝑑)))
26	25	imbi2d 340	. . . 4 ⊢ (𝑏 = 𝑑 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝑑) ≤ (𝐺‘𝑑))))
27		fveq2 6920	. . . . . 6 ⊢ (𝑏 = (𝑑 + 1) → (𝐹‘𝑏) = (𝐹‘(𝑑 + 1)))
28		fveq2 6920	. . . . . 6 ⊢ (𝑏 = (𝑑 + 1) → (𝐺‘𝑏) = (𝐺‘(𝑑 + 1)))
29	27, 28	breq12d 5179	. . . . 5 ⊢ (𝑏 = (𝑑 + 1) → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1))))
30	29	imbi2d 340	. . . 4 ⊢ (𝑏 = (𝑑 + 1) → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1)))))
31		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
32		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐺‘𝑏) = (𝐺‘𝐵))
33	31, 32	breq12d 5179	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝐵) ≤ (𝐺‘𝐵)))
34	33	imbi2d 340	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝐵) ≤ (𝐺‘𝐵))))
35		nnnn0 12560	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
36	35	faccld 14333	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℕ)
37	36	nnzd 12666	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℤ)
38	37	znegcld 12749	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → -(!‘𝐴) ∈ ℤ)
39		rpexpcl 14131	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
40	9, 38, 39	sylancr 586	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ⁺)
41	40	rpred 13099	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ)
42	41	leidd 11856	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ≤ (2↑-(!‘𝐴)))
43		nncn 12301	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
44	43	subidd 11635	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴 − 𝐴) = 0)
45	44	oveq2d 7464	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1 / 2)↑(𝐴 − 𝐴)) = ((1 / 2)↑0))
46		halfcn 12508	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℂ
47		exp0 14116	. . . . . . . . . 10 ⊢ ((1 / 2) ∈ ℂ → ((1 / 2)↑0) = 1)
48	46, 47	ax-mp 5	. . . . . . . . 9 ⊢ ((1 / 2)↑0) = 1
49	45, 48	eqtrdi 2796	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((1 / 2)↑(𝐴 − 𝐴)) = 1)
50	49	oveq2d 7464	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) = ((2↑-(!‘𝐴)) · 1))
51	40	rpcnd 13101	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℂ)
52	51	mulridd 11307	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · 1) = (2↑-(!‘𝐴)))
53	50, 52	eqtrd 2780	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) = (2↑-(!‘𝐴)))
54	42, 53	breqtrrd 5194	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))))
55		fveq2 6920	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
56	55	negeqd 11530	. . . . . . 7 ⊢ (𝑎 = 𝐴 → -(!‘𝑎) = -(!‘𝐴))
57	56	oveq2d 7464	. . . . . 6 ⊢ (𝑎 = 𝐴 → (2↑-(!‘𝑎)) = (2↑-(!‘𝐴)))
58		ovex 7481	. . . . . 6 ⊢ (2↑-(!‘𝐴)) ∈ V
59	57, 5, 58	fvmpt 7029	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = (2↑-(!‘𝐴)))
60		nnz 12660	. . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
61		uzid 12918	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ (ℤ_≥‘𝐴))
62		oveq1 7455	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (𝑐 − 𝐴) = (𝐴 − 𝐴))
63	62	oveq2d 7464	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝐴 − 𝐴)))
64	63	oveq2d 7464	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))))
65		aaliou3lem.a	. . . . . . 7 ⊢ 𝐺 = (𝑐 ∈ (ℤ_≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))))
66		ovex 7481	. . . . . . 7 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) ∈ V
67	64, 65, 66	fvmpt 7029	. . . . . 6 ⊢ (𝐴 ∈ (ℤ_≥‘𝐴) → (𝐺‘𝐴) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))))
68	60, 61, 67	3syl 18	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))))
69	54, 59, 68	3brtr4d 5198	. . . 4 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) ≤ (𝐺‘𝐴))
70		eluznn 12983	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℕ)
71	70	nnnn0d 12613	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℕ₀)
72	71	faccld 14333	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℕ)
73	72	nnzd 12666	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℤ)
74	73	znegcld 12749	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝑑) ∈ ℤ)
75		rpexpcl 14131	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑑) ∈ ℤ) → (2↑-(!‘𝑑)) ∈ ℝ⁺)
76	9, 74, 75	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝑑)) ∈ ℝ⁺)
77	76	rpred 13099	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝑑)) ∈ ℝ)
78	76	rpge0d 13103	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 0 ≤ (2↑-(!‘𝑑)))
79		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℕ)
80	79	nnnn0d 12613	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℕ₀)
81	80	faccld 14333	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝐴) ∈ ℕ)
82	81	nnzd 12666	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝐴) ∈ ℤ)
83	82	znegcld 12749	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝐴) ∈ ℤ)
84	9, 83, 39	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
85		halfre 12507	. . . . . . . . . . . . . . . 16 ⊢ (1 / 2) ∈ ℝ
86		halfgt0 12509	. . . . . . . . . . . . . . . 16 ⊢ 0 < (1 / 2)
87	85, 86	elrpii 13060	. . . . . . . . . . . . . . 15 ⊢ (1 / 2) ∈ ℝ⁺
88		eluzelz 12913	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → 𝑑 ∈ ℤ)
89		zsubcl 12685	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑑 − 𝐴) ∈ ℤ)
90	88, 60, 89	syl2anr 596	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 − 𝐴) ∈ ℤ)
91		rpexpcl 14131	. . . . . . . . . . . . . . 15 ⊢ (((1 / 2) ∈ ℝ⁺ ∧ (𝑑 − 𝐴) ∈ ℤ) → ((1 / 2)↑(𝑑 − 𝐴)) ∈ ℝ⁺)
92	87, 90, 91	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑(𝑑 − 𝐴)) ∈ ℝ⁺)
93	84, 92	rpmulcld 13115	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ⁺)
94	93	rpred 13099	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ)
95	77, 78, 94	jca31 514	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (((2↑-(!‘𝑑)) ∈ ℝ ∧ 0 ≤ (2↑-(!‘𝑑))) ∧ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ))
96	95	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (((2↑-(!‘𝑑)) ∈ ℝ ∧ 0 ≤ (2↑-(!‘𝑑))) ∧ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ))
97	88	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℤ)
98	74, 97	zmulcld 12753	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) ∈ ℤ)
99		rpexpcl 14131	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ⁺ ∧ (-(!‘𝑑) · 𝑑) ∈ ℤ) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ⁺)
100	9, 98, 99	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ⁺)
101	100	rpred 13099	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ)
102	100	rpge0d 13103	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 0 ≤ (2↑(-(!‘𝑑) · 𝑑)))
103	85	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (1 / 2) ∈ ℝ)
104	101, 102, 103	jca31 514	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (((2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ ∧ 0 ≤ (2↑(-(!‘𝑑) · 𝑑))) ∧ (1 / 2) ∈ ℝ))
105	104	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (((2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ ∧ 0 ≤ (2↑(-(!‘𝑑) · 𝑑))) ∧ (1 / 2) ∈ ℝ))
106		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
107		2re 12367	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
108		1le2 12502	. . . . . . . . . . . . 13 ⊢ 1 ≤ 2
109	72	nncnd 12309	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℂ)
110	97	zcnd 12748	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℂ)
111	109, 110	mulneg1d 11743	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) = -((!‘𝑑) · 𝑑))
112	72, 70	nnmulcld 12346	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((!‘𝑑) · 𝑑) ∈ ℕ)
113	112	nnge1d 12341	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 1 ≤ ((!‘𝑑) · 𝑑))
114		1re 11290	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
115	112	nnred 12308	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((!‘𝑑) · 𝑑) ∈ ℝ)
116		leneg 11793	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ ((!‘𝑑) · 𝑑) ∈ ℝ) → (1 ≤ ((!‘𝑑) · 𝑑) ↔ -((!‘𝑑) · 𝑑) ≤ -1))
117	114, 115, 116	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (1 ≤ ((!‘𝑑) · 𝑑) ↔ -((!‘𝑑) · 𝑑) ≤ -1))
118	113, 117	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -((!‘𝑑) · 𝑑) ≤ -1)
119	111, 118	eqbrtrd 5188	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) ≤ -1)
120		neg1z 12679	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℤ
121		eluz 12917	. . . . . . . . . . . . . . 15 ⊢ (((-(!‘𝑑) · 𝑑) ∈ ℤ ∧ -1 ∈ ℤ) → (-1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑)) ↔ (-(!‘𝑑) · 𝑑) ≤ -1))
122	98, 120, 121	sylancl 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑)) ↔ (-(!‘𝑑) · 𝑑) ≤ -1))
123	119, 122	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑)))
124		leexp2a 14222	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ -1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑))) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (2↑-1))
125	107, 108, 123, 124	mp3an12i 1465	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (2↑-1))
126		2cn 12368	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
127		expn1 14122	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ → (2↑-1) = (1 / 2))
128	126, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2↑-1) = (1 / 2)
129	125, 128	breqtrdi 5207	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2))
130	129	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2))
131		lemul12a 12152	. . . . . . . . . . 11 ⊢ (((((2↑-(!‘𝑑)) ∈ ℝ ∧ 0 ≤ (2↑-(!‘𝑑))) ∧ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ) ∧ (((2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ ∧ 0 ≤ (2↑(-(!‘𝑑) · 𝑑))) ∧ (1 / 2) ∈ ℝ)) → (((2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∧ (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2)) → ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))) ≤ (((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) · (1 / 2))))
132	131	3impia 1117	. . . . . . . . . 10 ⊢ (((((2↑-(!‘𝑑)) ∈ ℝ ∧ 0 ≤ (2↑-(!‘𝑑))) ∧ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ) ∧ (((2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ ∧ 0 ≤ (2↑(-(!‘𝑑) · 𝑑))) ∧ (1 / 2) ∈ ℝ) ∧ ((2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∧ (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2))) → ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))) ≤ (((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) · (1 / 2)))
133	96, 105, 106, 130, 132	syl112anc 1374	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))) ≤ (((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) · (1 / 2)))
134	133	ex 412	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) → ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))) ≤ (((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) · (1 / 2))))
135		facp1 14327	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ₀ → (!‘(𝑑 + 1)) = ((!‘𝑑) · (𝑑 + 1)))
136	71, 135	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘(𝑑 + 1)) = ((!‘𝑑) · (𝑑 + 1)))
137	136	negeqd 11530	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘(𝑑 + 1)) = -((!‘𝑑) · (𝑑 + 1)))
138		ax-1cn 11242	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
139		addcom 11476	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑑 + 1) = (1 + 𝑑))
140	110, 138, 139	sylancl 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) = (1 + 𝑑))
141	140	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (𝑑 + 1)) = (-(!‘𝑑) · (1 + 𝑑)))
142		peano2cn 11462	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℂ → (𝑑 + 1) ∈ ℂ)
143	110, 142	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) ∈ ℂ)
144	109, 143	mulneg1d 11743	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (𝑑 + 1)) = -((!‘𝑑) · (𝑑 + 1)))
145	74	zcnd 12748	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝑑) ∈ ℂ)
146		1cnd 11285	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 1 ∈ ℂ)
147	145, 146, 110	adddid 11314	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (1 + 𝑑)) = ((-(!‘𝑑) · 1) + (-(!‘𝑑) · 𝑑)))
148	145	mulridd 11307	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 1) = -(!‘𝑑))
149	148	oveq1d 7463	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((-(!‘𝑑) · 1) + (-(!‘𝑑) · 𝑑)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
150	147, 149	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (1 + 𝑑)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
151	141, 144, 150	3eqtr3d 2788	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -((!‘𝑑) · (𝑑 + 1)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
152	137, 151	eqtrd 2780	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘(𝑑 + 1)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
153	152	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘(𝑑 + 1))) = (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))))
154		2cnne0 12503	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
155		expaddz 14157	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (-(!‘𝑑) ∈ ℤ ∧ (-(!‘𝑑) · 𝑑) ∈ ℤ)) → (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
156	154, 155	mpan 689	. . . . . . . . . . 11 ⊢ ((-(!‘𝑑) ∈ ℤ ∧ (-(!‘𝑑) · 𝑑) ∈ ℤ) → (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
157	74, 98, 156	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
158	153, 157	eqtrd 2780	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘(𝑑 + 1))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
159	43	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℂ)
160	110, 146, 159	addsubd 11668	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝑑 + 1) − 𝐴) = ((𝑑 − 𝐴) + 1))
161	160	oveq2d 7464	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 + 1) − 𝐴)) = ((1 / 2)↑((𝑑 − 𝐴) + 1)))
162		uznn0sub 12942	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝑑 − 𝐴) ∈ ℕ₀)
163	162	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 − 𝐴) ∈ ℕ₀)
164		expp1 14119	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℂ ∧ (𝑑 − 𝐴) ∈ ℕ₀) → ((1 / 2)↑((𝑑 − 𝐴) + 1)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
165	46, 163, 164	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 − 𝐴) + 1)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
166	161, 165	eqtrd 2780	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 + 1) − 𝐴)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
167	166	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) = ((2↑-(!‘𝐴)) · (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2))))
168	84	rpcnd 13101	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℂ)
169	92	rpcnd 13101	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑(𝑑 − 𝐴)) ∈ ℂ)
170	46	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (1 / 2) ∈ ℂ)
171	168, 169, 170	mulassd 11313	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) · (1 / 2)) = ((2↑-(!‘𝐴)) · (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2))))
172	167, 171	eqtr4d 2783	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) = (((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) · (1 / 2)))
173	158, 172	breq12d 5179	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘(𝑑 + 1))) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) ↔ ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))) ≤ (((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) · (1 / 2))))
174	134, 173	sylibrd 259	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) → (2↑-(!‘(𝑑 + 1))) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴)))))
175		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (!‘𝑎) = (!‘𝑑))
176	175	negeqd 11530	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑑 → -(!‘𝑎) = -(!‘𝑑))
177	176	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑑)))
178		ovex 7481	. . . . . . . . . 10 ⊢ (2↑-(!‘𝑑)) ∈ V
179	177, 5, 178	fvmpt 7029	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (𝐹‘𝑑) = (2↑-(!‘𝑑)))
180	70, 179	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑑) = (2↑-(!‘𝑑)))
181		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑑 → (𝑐 − 𝐴) = (𝑑 − 𝐴))
182	181	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑑 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝑑 − 𝐴)))
183	182	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
184		ovex 7481	. . . . . . . . . 10 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ V
185	183, 65, 184	fvmpt 7029	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝐺‘𝑑) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
186	185	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑑) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
187	180, 186	breq12d 5179	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝑑) ≤ (𝐺‘𝑑) ↔ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))))
188	70	peano2nnd 12310	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) ∈ ℕ)
189		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑑 + 1) → (!‘𝑎) = (!‘(𝑑 + 1)))
190	189	negeqd 11530	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑑 + 1) → -(!‘𝑎) = -(!‘(𝑑 + 1)))
191	190	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑎 = (𝑑 + 1) → (2↑-(!‘𝑎)) = (2↑-(!‘(𝑑 + 1))))
192		ovex 7481	. . . . . . . . . 10 ⊢ (2↑-(!‘(𝑑 + 1))) ∈ V
193	191, 5, 192	fvmpt 7029	. . . . . . . . 9 ⊢ ((𝑑 + 1) ∈ ℕ → (𝐹‘(𝑑 + 1)) = (2↑-(!‘(𝑑 + 1))))
194	188, 193	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐹‘(𝑑 + 1)) = (2↑-(!‘(𝑑 + 1))))
195		peano2uz 12966	. . . . . . . . . 10 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝑑 + 1) ∈ (ℤ_≥‘𝐴))
196		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑑 + 1) → (𝑐 − 𝐴) = ((𝑑 + 1) − 𝐴))
197	196	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑑 + 1) → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑((𝑑 + 1) − 𝐴)))
198	197	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑑 + 1) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
199		ovex 7481	. . . . . . . . . . 11 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) ∈ V
200	198, 65, 199	fvmpt 7029	. . . . . . . . . 10 ⊢ ((𝑑 + 1) ∈ (ℤ_≥‘𝐴) → (𝐺‘(𝑑 + 1)) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
201	195, 200	syl 17	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝐺‘(𝑑 + 1)) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
202	201	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐺‘(𝑑 + 1)) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
203	194, 202	breq12d 5179	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1)) ↔ (2↑-(!‘(𝑑 + 1))) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴)))))
204	174, 187, 203	3imtr4d 294	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝑑) ≤ (𝐺‘𝑑) → (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1))))
205	204	expcom 413	. . . . 5 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝐴 ∈ ℕ → ((𝐹‘𝑑) ≤ (𝐺‘𝑑) → (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1)))))
206	205	a2d 29	. . . 4 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → ((𝐴 ∈ ℕ → (𝐹‘𝑑) ≤ (𝐺‘𝑑)) → (𝐴 ∈ ℕ → (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1)))))
207	22, 26, 30, 34, 69, 206	uzind4i 12975	. . 3 ⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (𝐴 ∈ ℕ → (𝐹‘𝐵) ≤ (𝐺‘𝐵)))
208	207	impcom 407	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ≤ (𝐺‘𝐵))
209		0xr 11337	. . 3 ⊢ 0 ∈ ℝ^*
210	65	aaliou3lem1 26402	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ)
211		elioc2 13470	. . 3 ⊢ ((0 ∈ ℝ^* ∧ (𝐺‘𝐵) ∈ ℝ) → ((𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵)) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ 0 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ (𝐺‘𝐵))))
212	209, 210, 211	sylancr 586	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵)) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ 0 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ (𝐺‘𝐵))))
213	17, 18, 208, 212	mpbir3and 1342	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 (,]cioc 13408 ↑cexp 14112 !cfa 14322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ioc 13412 df-seq 14053 df-exp 14113 df-fac 14323

This theorem is referenced by: aaliou3lem3 26404

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