aaliou3lem2 - Metamath Proof Explorer

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Theorem aaliou3lem2 25825

Description: Lemma for aaliou3 25833. (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 12889	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 𝐵 ∈ ℕ)
2		fveq2 6881	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (!‘𝑎) = (!‘𝐵))
3	2	negeqd 11441	. . . . . . 7 ⊢ (𝑎 = 𝐵 → -(!‘𝑎) = -(!‘𝐵))
4	3	oveq2d 7412	. . . . . 6 ⊢ (𝑎 = 𝐵 → (2↑-(!‘𝑎)) = (2↑-(!‘𝐵)))
5		aaliou3lem.b	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
6		ovex 7429	. . . . . 6 ⊢ (2↑-(!‘𝐵)) ∈ V
7	4, 5, 6	fvmpt 6987	. . . . 5 ⊢ (𝐵 ∈ ℕ → (𝐹‘𝐵) = (2↑-(!‘𝐵)))
8	1, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) = (2↑-(!‘𝐵)))
9		2rp 12966	. . . . 5 ⊢ 2 ∈ ℝ⁺
10	1	nnnn0d 12519	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 𝐵 ∈ ℕ₀)
11	10	faccld 14231	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (!‘𝐵) ∈ ℕ)
12	11	nnzd 12572	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (!‘𝐵) ∈ ℤ)
13	12	znegcld 12655	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → -(!‘𝐵) ∈ ℤ)
14		rpexpcl 14033	. . . . 5 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐵) ∈ ℤ) → (2↑-(!‘𝐵)) ∈ ℝ⁺)
15	9, 13, 14	sylancr 588	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐵)) ∈ ℝ⁺)
16	8, 15	eqeltrd 2834	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ ℝ⁺)
17	16	rpred 13003	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ ℝ)
18	16	rpgt0d 13006	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 0 < (𝐹‘𝐵))
19		fveq2 6881	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝐹‘𝑏) = (𝐹‘𝐴))
20		fveq2 6881	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝐺‘𝑏) = (𝐺‘𝐴))
21	19, 20	breq12d 5157	. . . . 5 ⊢ (𝑏 = 𝐴 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝐴) ≤ (𝐺‘𝐴)))
22	21	imbi2d 341	. . . 4 ⊢ (𝑏 = 𝐴 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝐴) ≤ (𝐺‘𝐴))))
23		fveq2 6881	. . . . . 6 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
24		fveq2 6881	. . . . . 6 ⊢ (𝑏 = 𝑑 → (𝐺‘𝑏) = (𝐺‘𝑑))
25	23, 24	breq12d 5157	. . . . 5 ⊢ (𝑏 = 𝑑 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝑑) ≤ (𝐺‘𝑑)))
26	25	imbi2d 341	. . . 4 ⊢ (𝑏 = 𝑑 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝑑) ≤ (𝐺‘𝑑))))
27		fveq2 6881	. . . . . 6 ⊢ (𝑏 = (𝑑 + 1) → (𝐹‘𝑏) = (𝐹‘(𝑑 + 1)))
28		fveq2 6881	. . . . . 6 ⊢ (𝑏 = (𝑑 + 1) → (𝐺‘𝑏) = (𝐺‘(𝑑 + 1)))
29	27, 28	breq12d 5157	. . . . 5 ⊢ (𝑏 = (𝑑 + 1) → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1))))
30	29	imbi2d 341	. . . 4 ⊢ (𝑏 = (𝑑 + 1) → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1)))))
31		fveq2 6881	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
32		fveq2 6881	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐺‘𝑏) = (𝐺‘𝐵))
33	31, 32	breq12d 5157	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝐵) ≤ (𝐺‘𝐵)))
34	33	imbi2d 341	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝐵) ≤ (𝐺‘𝐵))))
35		nnnn0 12466	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
36	35	faccld 14231	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℕ)
37	36	nnzd 12572	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℤ)
38	37	znegcld 12655	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → -(!‘𝐴) ∈ ℤ)
39		rpexpcl 14033	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
40	9, 38, 39	sylancr 588	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ⁺)
41	40	rpred 13003	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ)
42	41	leidd 11767	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ≤ (2↑-(!‘𝐴)))
43		nncn 12207	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
44	43	subidd 11546	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴 − 𝐴) = 0)
45	44	oveq2d 7412	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1 / 2)↑(𝐴 − 𝐴)) = ((1 / 2)↑0))
46		halfcn 12414	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℂ
47		exp0 14018	. . . . . . . . . 10 ⊢ ((1 / 2) ∈ ℂ → ((1 / 2)↑0) = 1)
48	46, 47	ax-mp 5	. . . . . . . . 9 ⊢ ((1 / 2)↑0) = 1
49	45, 48	eqtrdi 2789	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((1 / 2)↑(𝐴 − 𝐴)) = 1)
50	49	oveq2d 7412	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) = ((2↑-(!‘𝐴)) · 1))
51	40	rpcnd 13005	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℂ)
52	51	mulridd 11218	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · 1) = (2↑-(!‘𝐴)))
53	50, 52	eqtrd 2773	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) = (2↑-(!‘𝐴)))
54	42, 53	breqtrrd 5172	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))))
55		fveq2 6881	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
56	55	negeqd 11441	. . . . . . 7 ⊢ (𝑎 = 𝐴 → -(!‘𝑎) = -(!‘𝐴))
57	56	oveq2d 7412	. . . . . 6 ⊢ (𝑎 = 𝐴 → (2↑-(!‘𝑎)) = (2↑-(!‘𝐴)))
58		ovex 7429	. . . . . 6 ⊢ (2↑-(!‘𝐴)) ∈ V
59	57, 5, 58	fvmpt 6987	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = (2↑-(!‘𝐴)))
60		nnz 12566	. . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
61		uzid 12824	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ (ℤ_≥‘𝐴))
62		oveq1 7403	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (𝑐 − 𝐴) = (𝐴 − 𝐴))
63	62	oveq2d 7412	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝐴 − 𝐴)))
64	63	oveq2d 7412	. . . . . . 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 7429	. . . . . . 7 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) ∈ V
67	64, 65, 66	fvmpt 6987	. . . . . 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 5176	. . . 4 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) ≤ (𝐺‘𝐴))
70		eluznn 12889	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℕ)
71	70	nnnn0d 12519	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℕ₀)
72	71	faccld 14231	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℕ)
73	72	nnzd 12572	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℤ)
74	73	znegcld 12655	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝑑) ∈ ℤ)
75		rpexpcl 14033	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑑) ∈ ℤ) → (2↑-(!‘𝑑)) ∈ ℝ⁺)
76	9, 74, 75	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝑑)) ∈ ℝ⁺)
77	76	rpred 13003	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝑑)) ∈ ℝ)
78	76	rpge0d 13007	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 0 ≤ (2↑-(!‘𝑑)))
79		simpl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℕ)
80	79	nnnn0d 12519	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℕ₀)
81	80	faccld 14231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝐴) ∈ ℕ)
82	81	nnzd 12572	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝐴) ∈ ℤ)
83	82	znegcld 12655	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝐴) ∈ ℤ)
84	9, 83, 39	sylancr 588	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
85		halfre 12413	. . . . . . . . . . . . . . . 16 ⊢ (1 / 2) ∈ ℝ
86		halfgt0 12415	. . . . . . . . . . . . . . . 16 ⊢ 0 < (1 / 2)
87	85, 86	elrpii 12964	. . . . . . . . . . . . . . 15 ⊢ (1 / 2) ∈ ℝ⁺
88		eluzelz 12819	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → 𝑑 ∈ ℤ)
89		zsubcl 12591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑑 − 𝐴) ∈ ℤ)
90	88, 60, 89	syl2anr 598	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 − 𝐴) ∈ ℤ)
91		rpexpcl 14033	. . . . . . . . . . . . . . 15 ⊢ (((1 / 2) ∈ ℝ⁺ ∧ (𝑑 − 𝐴) ∈ ℤ) → ((1 / 2)↑(𝑑 − 𝐴)) ∈ ℝ⁺)
92	87, 90, 91	sylancr 588	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑(𝑑 − 𝐴)) ∈ ℝ⁺)
93	84, 92	rpmulcld 13019	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ⁺)
94	93	rpred 13003	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ)
95	77, 78, 94	jca31 516	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (((2↑-(!‘𝑑)) ∈ ℝ ∧ 0 ≤ (2↑-(!‘𝑑))) ∧ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ))
96	95	adantr 482	. . . . . . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℤ)
98	74, 97	zmulcld 12659	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) ∈ ℤ)
99		rpexpcl 14033	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ⁺ ∧ (-(!‘𝑑) · 𝑑) ∈ ℤ) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ⁺)
100	9, 98, 99	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ⁺)
101	100	rpred 13003	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ)
102	100	rpge0d 13007	. . . . . . . . . . . 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 516	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (((2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ ∧ 0 ≤ (2↑(-(!‘𝑑) · 𝑑))) ∧ (1 / 2) ∈ ℝ))
105	104	adantr 482	. . . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
107		2re 12273	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
108		1le2 12408	. . . . . . . . . . . . 13 ⊢ 1 ≤ 2
109	72	nncnd 12215	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℂ)
110	97	zcnd 12654	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℂ)
111	109, 110	mulneg1d 11654	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) = -((!‘𝑑) · 𝑑))
112	72, 70	nnmulcld 12252	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((!‘𝑑) · 𝑑) ∈ ℕ)
113	112	nnge1d 12247	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 1 ≤ ((!‘𝑑) · 𝑑))
114		1re 11201	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
115	112	nnred 12214	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((!‘𝑑) · 𝑑) ∈ ℝ)
116		leneg 11704	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ ((!‘𝑑) · 𝑑) ∈ ℝ) → (1 ≤ ((!‘𝑑) · 𝑑) ↔ -((!‘𝑑) · 𝑑) ≤ -1))
117	114, 115, 116	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (1 ≤ ((!‘𝑑) · 𝑑) ↔ -((!‘𝑑) · 𝑑) ≤ -1))
118	113, 117	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -((!‘𝑑) · 𝑑) ≤ -1)
119	111, 118	eqbrtrd 5166	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) ≤ -1)
120		neg1z 12585	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℤ
121		eluz 12823	. . . . . . . . . . . . . . 15 ⊢ (((-(!‘𝑑) · 𝑑) ∈ ℤ ∧ -1 ∈ ℤ) → (-1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑)) ↔ (-(!‘𝑑) · 𝑑) ≤ -1))
122	98, 120, 121	sylancl 587	. . . . . . . . . . . . . 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 14124	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ -1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑))) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (2↑-1))
125	107, 108, 123, 124	mp3an12i 1466	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (2↑-1))
126		2cn 12274	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
127		expn1 14024	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ → (2↑-1) = (1 / 2))
128	126, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2↑-1) = (1 / 2)
129	125, 128	breqtrdi 5185	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2))
130	129	adantr 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2))
131		lemul12a 12059	. . . . . . . . . . 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 1118	. . . . . . . . . 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 1375	. . . . . . . . 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 414	. . . . . . . 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 14225	. . . . . . . . . . . . . 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 11441	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘(𝑑 + 1)) = -((!‘𝑑) · (𝑑 + 1)))
138		ax-1cn 11155	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
139		addcom 11387	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑑 + 1) = (1 + 𝑑))
140	110, 138, 139	sylancl 587	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) = (1 + 𝑑))
141	140	oveq2d 7412	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (𝑑 + 1)) = (-(!‘𝑑) · (1 + 𝑑)))
142		peano2cn 11373	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℂ → (𝑑 + 1) ∈ ℂ)
143	110, 142	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) ∈ ℂ)
144	109, 143	mulneg1d 11654	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (𝑑 + 1)) = -((!‘𝑑) · (𝑑 + 1)))
145	74	zcnd 12654	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝑑) ∈ ℂ)
146		1cnd 11196	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 1 ∈ ℂ)
147	145, 146, 110	adddid 11225	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (1 + 𝑑)) = ((-(!‘𝑑) · 1) + (-(!‘𝑑) · 𝑑)))
148	145	mulridd 11218	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 1) = -(!‘𝑑))
149	148	oveq1d 7411	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((-(!‘𝑑) · 1) + (-(!‘𝑑) · 𝑑)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
150	147, 149	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (1 + 𝑑)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
151	141, 144, 150	3eqtr3d 2781	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -((!‘𝑑) · (𝑑 + 1)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
152	137, 151	eqtrd 2773	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘(𝑑 + 1)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
153	152	oveq2d 7412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘(𝑑 + 1))) = (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))))
154		2cnne0 12409	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
155		expaddz 14059	. . . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
158	153, 157	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘(𝑑 + 1))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
159	43	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℂ)
160	110, 146, 159	addsubd 11579	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝑑 + 1) − 𝐴) = ((𝑑 − 𝐴) + 1))
161	160	oveq2d 7412	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 + 1) − 𝐴)) = ((1 / 2)↑((𝑑 − 𝐴) + 1)))
162		uznn0sub 12848	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝑑 − 𝐴) ∈ ℕ₀)
163	162	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 − 𝐴) ∈ ℕ₀)
164		expp1 14021	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℂ ∧ (𝑑 − 𝐴) ∈ ℕ₀) → ((1 / 2)↑((𝑑 − 𝐴) + 1)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
165	46, 163, 164	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 − 𝐴) + 1)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
166	161, 165	eqtrd 2773	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 + 1) − 𝐴)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
167	166	oveq2d 7412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) = ((2↑-(!‘𝐴)) · (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2))))
168	84	rpcnd 13005	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℂ)
169	92	rpcnd 13005	. . . . . . . . . . 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 11224	. . . . . . . . . 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 2776	. . . . . . . . 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 5157	. . . . . . . 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 6881	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (!‘𝑎) = (!‘𝑑))
176	175	negeqd 11441	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑑 → -(!‘𝑎) = -(!‘𝑑))
177	176	oveq2d 7412	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑑)))
178		ovex 7429	. . . . . . . . . 10 ⊢ (2↑-(!‘𝑑)) ∈ V
179	177, 5, 178	fvmpt 6987	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (𝐹‘𝑑) = (2↑-(!‘𝑑)))
180	70, 179	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑑) = (2↑-(!‘𝑑)))
181		oveq1 7403	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑑 → (𝑐 − 𝐴) = (𝑑 − 𝐴))
182	181	oveq2d 7412	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑑 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝑑 − 𝐴)))
183	182	oveq2d 7412	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
184		ovex 7429	. . . . . . . . . 10 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ V
185	183, 65, 184	fvmpt 6987	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝐺‘𝑑) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
186	185	adantl 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑑) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
187	180, 186	breq12d 5157	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝑑) ≤ (𝐺‘𝑑) ↔ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))))
188	70	peano2nnd 12216	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) ∈ ℕ)
189		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑑 + 1) → (!‘𝑎) = (!‘(𝑑 + 1)))
190	189	negeqd 11441	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑑 + 1) → -(!‘𝑎) = -(!‘(𝑑 + 1)))
191	190	oveq2d 7412	. . . . . . . . . 10 ⊢ (𝑎 = (𝑑 + 1) → (2↑-(!‘𝑎)) = (2↑-(!‘(𝑑 + 1))))
192		ovex 7429	. . . . . . . . . 10 ⊢ (2↑-(!‘(𝑑 + 1))) ∈ V
193	191, 5, 192	fvmpt 6987	. . . . . . . . 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 12872	. . . . . . . . . 10 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝑑 + 1) ∈ (ℤ_≥‘𝐴))
196		oveq1 7403	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑑 + 1) → (𝑐 − 𝐴) = ((𝑑 + 1) − 𝐴))
197	196	oveq2d 7412	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑑 + 1) → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑((𝑑 + 1) − 𝐴)))
198	197	oveq2d 7412	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑑 + 1) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
199		ovex 7429	. . . . . . . . . . 11 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) ∈ V
200	198, 65, 199	fvmpt 6987	. . . . . . . . . 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 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐺‘(𝑑 + 1)) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
203	194, 202	breq12d 5157	. . . . . . 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 415	. . . . 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 12881	. . 3 ⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (𝐴 ∈ ℕ → (𝐹‘𝐵) ≤ (𝐺‘𝐵)))
208	207	impcom 409	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ≤ (𝐺‘𝐵))
209		0xr 11248	. . 3 ⊢ 0 ∈ ℝ^*
210	65	aaliou3lem1 25824	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ)
211		elioc2 13374	. . 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 588	. 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 1343	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 ℝcr 11096 0cc0 11097 1c1 11098 + caddc 11100 · cmul 11102 ℝ^*cxr 11234 < clt 11235 ≤ cle 11236 − cmin 11431 -cneg 11432 / cdiv 11858 ℕcn 12199 2c2 12254 ℕ₀cn0 12459 ℤcz 12545 ℤ_≥cuz 12809 ℝ⁺crp 12961 (,]cioc 13312 ↑cexp 14014 !cfa 14220

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-n0 12460 df-z 12546 df-uz 12810 df-rp 12962 df-ioc 13316 df-seq 13954 df-exp 14015 df-fac 14221

This theorem is referenced by: aaliou3lem3 25826

W3C validator