aaliou3lem2 - Metamath Proof Explorer

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Theorem aaliou3lem2 25031

Description: Lemma for aaliou3 25039. (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 12351	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 𝐵 ∈ ℕ)
2		fveq2 6659	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (!‘𝑎) = (!‘𝐵))
3	2	negeqd 10911	. . . . . . 7 ⊢ (𝑎 = 𝐵 → -(!‘𝑎) = -(!‘𝐵))
4	3	oveq2d 7167	. . . . . 6 ⊢ (𝑎 = 𝐵 → (2↑-(!‘𝑎)) = (2↑-(!‘𝐵)))
5		aaliou3lem.b	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
6		ovex 7184	. . . . . 6 ⊢ (2↑-(!‘𝐵)) ∈ V
7	4, 5, 6	fvmpt 6760	. . . . 5 ⊢ (𝐵 ∈ ℕ → (𝐹‘𝐵) = (2↑-(!‘𝐵)))
8	1, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) = (2↑-(!‘𝐵)))
9		2rp 12428	. . . . 5 ⊢ 2 ∈ ℝ⁺
10	1	nnnn0d 11987	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 𝐵 ∈ ℕ₀)
11	10	faccld 13687	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (!‘𝐵) ∈ ℕ)
12	11	nnzd 12118	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (!‘𝐵) ∈ ℤ)
13	12	znegcld 12121	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → -(!‘𝐵) ∈ ℤ)
14		rpexpcl 13491	. . . . 5 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐵) ∈ ℤ) → (2↑-(!‘𝐵)) ∈ ℝ⁺)
15	9, 13, 14	sylancr 591	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐵)) ∈ ℝ⁺)
16	8, 15	eqeltrd 2853	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ ℝ⁺)
17	16	rpred 12465	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ ℝ)
18	16	rpgt0d 12468	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → 0 < (𝐹‘𝐵))
19		fveq2 6659	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝐹‘𝑏) = (𝐹‘𝐴))
20		fveq2 6659	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝐺‘𝑏) = (𝐺‘𝐴))
21	19, 20	breq12d 5046	. . . . 5 ⊢ (𝑏 = 𝐴 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝐴) ≤ (𝐺‘𝐴)))
22	21	imbi2d 345	. . . 4 ⊢ (𝑏 = 𝐴 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝐴) ≤ (𝐺‘𝐴))))
23		fveq2 6659	. . . . . 6 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
24		fveq2 6659	. . . . . 6 ⊢ (𝑏 = 𝑑 → (𝐺‘𝑏) = (𝐺‘𝑑))
25	23, 24	breq12d 5046	. . . . 5 ⊢ (𝑏 = 𝑑 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝑑) ≤ (𝐺‘𝑑)))
26	25	imbi2d 345	. . . 4 ⊢ (𝑏 = 𝑑 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝑑) ≤ (𝐺‘𝑑))))
27		fveq2 6659	. . . . . 6 ⊢ (𝑏 = (𝑑 + 1) → (𝐹‘𝑏) = (𝐹‘(𝑑 + 1)))
28		fveq2 6659	. . . . . 6 ⊢ (𝑏 = (𝑑 + 1) → (𝐺‘𝑏) = (𝐺‘(𝑑 + 1)))
29	27, 28	breq12d 5046	. . . . 5 ⊢ (𝑏 = (𝑑 + 1) → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1))))
30	29	imbi2d 345	. . . 4 ⊢ (𝑏 = (𝑑 + 1) → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1)))))
31		fveq2 6659	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
32		fveq2 6659	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐺‘𝑏) = (𝐺‘𝐵))
33	31, 32	breq12d 5046	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑏) ≤ (𝐺‘𝑏) ↔ (𝐹‘𝐵) ≤ (𝐺‘𝐵)))
34	33	imbi2d 345	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ ℕ → (𝐹‘𝑏) ≤ (𝐺‘𝑏)) ↔ (𝐴 ∈ ℕ → (𝐹‘𝐵) ≤ (𝐺‘𝐵))))
35		nnnn0 11934	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
36	35	faccld 13687	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℕ)
37	36	nnzd 12118	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℤ)
38	37	znegcld 12121	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → -(!‘𝐴) ∈ ℤ)
39		rpexpcl 13491	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
40	9, 38, 39	sylancr 591	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ⁺)
41	40	rpred 12465	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ)
42	41	leidd 11237	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ≤ (2↑-(!‘𝐴)))
43		nncn 11675	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
44	43	subidd 11016	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴 − 𝐴) = 0)
45	44	oveq2d 7167	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1 / 2)↑(𝐴 − 𝐴)) = ((1 / 2)↑0))
46		halfcn 11882	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℂ
47		exp0 13476	. . . . . . . . . 10 ⊢ ((1 / 2) ∈ ℂ → ((1 / 2)↑0) = 1)
48	46, 47	ax-mp 5	. . . . . . . . 9 ⊢ ((1 / 2)↑0) = 1
49	45, 48	eqtrdi 2810	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((1 / 2)↑(𝐴 − 𝐴)) = 1)
50	49	oveq2d 7167	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) = ((2↑-(!‘𝐴)) · 1))
51	40	rpcnd 12467	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℂ)
52	51	mulid1d 10689	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · 1) = (2↑-(!‘𝐴)))
53	50, 52	eqtrd 2794	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) = (2↑-(!‘𝐴)))
54	42, 53	breqtrrd 5061	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))))
55		fveq2 6659	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
56	55	negeqd 10911	. . . . . . 7 ⊢ (𝑎 = 𝐴 → -(!‘𝑎) = -(!‘𝐴))
57	56	oveq2d 7167	. . . . . 6 ⊢ (𝑎 = 𝐴 → (2↑-(!‘𝑎)) = (2↑-(!‘𝐴)))
58		ovex 7184	. . . . . 6 ⊢ (2↑-(!‘𝐴)) ∈ V
59	57, 5, 58	fvmpt 6760	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = (2↑-(!‘𝐴)))
60		nnz 12036	. . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
61		uzid 12290	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ (ℤ_≥‘𝐴))
62		oveq1 7158	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (𝑐 − 𝐴) = (𝐴 − 𝐴))
63	62	oveq2d 7167	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝐴 − 𝐴)))
64	63	oveq2d 7167	. . . . . . 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 7184	. . . . . . 7 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐴 − 𝐴))) ∈ V
67	64, 65, 66	fvmpt 6760	. . . . . 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 5065	. . . 4 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) ≤ (𝐺‘𝐴))
70		eluznn 12351	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℕ)
71	70	nnnn0d 11987	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℕ₀)
72	71	faccld 13687	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℕ)
73	72	nnzd 12118	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℤ)
74	73	znegcld 12121	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝑑) ∈ ℤ)
75		rpexpcl 13491	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑑) ∈ ℤ) → (2↑-(!‘𝑑)) ∈ ℝ⁺)
76	9, 74, 75	sylancr 591	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝑑)) ∈ ℝ⁺)
77	76	rpred 12465	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝑑)) ∈ ℝ)
78	76	rpge0d 12469	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 0 ≤ (2↑-(!‘𝑑)))
79		simpl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℕ)
80	79	nnnn0d 11987	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℕ₀)
81	80	faccld 13687	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝐴) ∈ ℕ)
82	81	nnzd 12118	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝐴) ∈ ℤ)
83	82	znegcld 12121	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝐴) ∈ ℤ)
84	9, 83, 39	sylancr 591	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
85		halfre 11881	. . . . . . . . . . . . . . . 16 ⊢ (1 / 2) ∈ ℝ
86		halfgt0 11883	. . . . . . . . . . . . . . . 16 ⊢ 0 < (1 / 2)
87	85, 86	elrpii 12426	. . . . . . . . . . . . . . 15 ⊢ (1 / 2) ∈ ℝ⁺
88		eluzelz 12285	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → 𝑑 ∈ ℤ)
89		zsubcl 12056	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑑 − 𝐴) ∈ ℤ)
90	88, 60, 89	syl2anr 600	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 − 𝐴) ∈ ℤ)
91		rpexpcl 13491	. . . . . . . . . . . . . . 15 ⊢ (((1 / 2) ∈ ℝ⁺ ∧ (𝑑 − 𝐴) ∈ ℤ) → ((1 / 2)↑(𝑑 − 𝐴)) ∈ ℝ⁺)
92	87, 90, 91	sylancr 591	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑(𝑑 − 𝐴)) ∈ ℝ⁺)
93	84, 92	rpmulcld 12481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ⁺)
94	93	rpred 12465	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ)
95	77, 78, 94	jca31 519	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (((2↑-(!‘𝑑)) ∈ ℝ ∧ 0 ≤ (2↑-(!‘𝑑))) ∧ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ ℝ))
96	95	adantr 485	. . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℤ)
98	74, 97	zmulcld 12125	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) ∈ ℤ)
99		rpexpcl 13491	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ⁺ ∧ (-(!‘𝑑) · 𝑑) ∈ ℤ) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ⁺)
100	9, 98, 99	sylancr 591	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ⁺)
101	100	rpred 12465	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ)
102	100	rpge0d 12469	. . . . . . . . . . . 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 519	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (((2↑(-(!‘𝑑) · 𝑑)) ∈ ℝ ∧ 0 ≤ (2↑(-(!‘𝑑) · 𝑑))) ∧ (1 / 2) ∈ ℝ))
105	104	adantr 485	. . . . . . . . . 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 489	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
107		2re 11741	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
108		1le2 11876	. . . . . . . . . . . . 13 ⊢ 1 ≤ 2
109	72	nncnd 11683	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (!‘𝑑) ∈ ℂ)
110	97	zcnd 12120	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝑑 ∈ ℂ)
111	109, 110	mulneg1d 11124	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) = -((!‘𝑑) · 𝑑))
112	72, 70	nnmulcld 11720	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((!‘𝑑) · 𝑑) ∈ ℕ)
113	112	nnge1d 11715	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 1 ≤ ((!‘𝑑) · 𝑑))
114		1re 10672	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
115	112	nnred 11682	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((!‘𝑑) · 𝑑) ∈ ℝ)
116		leneg 11174	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ ((!‘𝑑) · 𝑑) ∈ ℝ) → (1 ≤ ((!‘𝑑) · 𝑑) ↔ -((!‘𝑑) · 𝑑) ≤ -1))
117	114, 115, 116	sylancr 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (1 ≤ ((!‘𝑑) · 𝑑) ↔ -((!‘𝑑) · 𝑑) ≤ -1))
118	113, 117	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -((!‘𝑑) · 𝑑) ≤ -1)
119	111, 118	eqbrtrd 5055	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 𝑑) ≤ -1)
120		neg1z 12050	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℤ
121		eluz 12289	. . . . . . . . . . . . . . 15 ⊢ (((-(!‘𝑑) · 𝑑) ∈ ℤ ∧ -1 ∈ ℤ) → (-1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑)) ↔ (-(!‘𝑑) · 𝑑) ≤ -1))
122	98, 120, 121	sylancl 590	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑)) ↔ (-(!‘𝑑) · 𝑑) ≤ -1))
123	119, 122	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑)))
124		leexp2a 13579	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ -1 ∈ (ℤ_≥‘(-(!‘𝑑) · 𝑑))) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (2↑-1))
125	107, 108, 123, 124	mp3an12i 1463	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (2↑-1))
126		2cn 11742	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
127		expn1 13482	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ → (2↑-1) = (1 / 2))
128	126, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2↑-1) = (1 / 2)
129	125, 128	breqtrdi 5074	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2))
130	129	adantr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) ∧ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))) → (2↑(-(!‘𝑑) · 𝑑)) ≤ (1 / 2))
131		lemul12a 11529	. . . . . . . . . . 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 1115	. . . . . . . . . 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 1372	. . . . . . . . 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 417	. . . . . . . 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 13681	. . . . . . . . . . . . . 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 10911	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘(𝑑 + 1)) = -((!‘𝑑) · (𝑑 + 1)))
138		ax-1cn 10626	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
139		addcom 10857	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑑 + 1) = (1 + 𝑑))
140	110, 138, 139	sylancl 590	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) = (1 + 𝑑))
141	140	oveq2d 7167	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (𝑑 + 1)) = (-(!‘𝑑) · (1 + 𝑑)))
142		peano2cn 10843	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℂ → (𝑑 + 1) ∈ ℂ)
143	110, 142	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) ∈ ℂ)
144	109, 143	mulneg1d 11124	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (𝑑 + 1)) = -((!‘𝑑) · (𝑑 + 1)))
145	74	zcnd 12120	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘𝑑) ∈ ℂ)
146		1cnd 10667	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 1 ∈ ℂ)
147	145, 146, 110	adddid 10696	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (1 + 𝑑)) = ((-(!‘𝑑) · 1) + (-(!‘𝑑) · 𝑑)))
148	145	mulid1d 10689	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · 1) = -(!‘𝑑))
149	148	oveq1d 7166	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((-(!‘𝑑) · 1) + (-(!‘𝑑) · 𝑑)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
150	147, 149	eqtrd 2794	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (-(!‘𝑑) · (1 + 𝑑)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
151	141, 144, 150	3eqtr3d 2802	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -((!‘𝑑) · (𝑑 + 1)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
152	137, 151	eqtrd 2794	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → -(!‘(𝑑 + 1)) = (-(!‘𝑑) + (-(!‘𝑑) · 𝑑)))
153	152	oveq2d 7167	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘(𝑑 + 1))) = (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))))
154		2cnne0 11877	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
155		expaddz 13516	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (-(!‘𝑑) ∈ ℤ ∧ (-(!‘𝑑) · 𝑑) ∈ ℤ)) → (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
156	154, 155	mpan 690	. . . . . . . . . . 11 ⊢ ((-(!‘𝑑) ∈ ℤ ∧ (-(!‘𝑑) · 𝑑) ∈ ℤ) → (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
157	74, 98, 156	syl2anc 588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑(-(!‘𝑑) + (-(!‘𝑑) · 𝑑))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
158	153, 157	eqtrd 2794	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘(𝑑 + 1))) = ((2↑-(!‘𝑑)) · (2↑(-(!‘𝑑) · 𝑑))))
159	43	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → 𝐴 ∈ ℂ)
160	110, 146, 159	addsubd 11049	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝑑 + 1) − 𝐴) = ((𝑑 − 𝐴) + 1))
161	160	oveq2d 7167	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 + 1) − 𝐴)) = ((1 / 2)↑((𝑑 − 𝐴) + 1)))
162		uznn0sub 12310	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝑑 − 𝐴) ∈ ℕ₀)
163	162	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 − 𝐴) ∈ ℕ₀)
164		expp1 13479	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℂ ∧ (𝑑 − 𝐴) ∈ ℕ₀) → ((1 / 2)↑((𝑑 − 𝐴) + 1)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
165	46, 163, 164	sylancr 591	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 − 𝐴) + 1)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
166	161, 165	eqtrd 2794	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((1 / 2)↑((𝑑 + 1) − 𝐴)) = (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2)))
167	166	oveq2d 7167	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) = ((2↑-(!‘𝐴)) · (((1 / 2)↑(𝑑 − 𝐴)) · (1 / 2))))
168	84	rpcnd 12467	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℂ)
169	92	rpcnd 12467	. . . . . . . . . . 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 10695	. . . . . . . . . 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 2797	. . . . . . . . 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 5046	. . . . . . . 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 262	. . . . . . 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 6659	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (!‘𝑎) = (!‘𝑑))
176	175	negeqd 10911	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑑 → -(!‘𝑎) = -(!‘𝑑))
177	176	oveq2d 7167	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑑)))
178		ovex 7184	. . . . . . . . . 10 ⊢ (2↑-(!‘𝑑)) ∈ V
179	177, 5, 178	fvmpt 6760	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (𝐹‘𝑑) = (2↑-(!‘𝑑)))
180	70, 179	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑑) = (2↑-(!‘𝑑)))
181		oveq1 7158	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑑 → (𝑐 − 𝐴) = (𝑑 − 𝐴))
182	181	oveq2d 7167	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑑 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝑑 − 𝐴)))
183	182	oveq2d 7167	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
184		ovex 7184	. . . . . . . . . 10 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))) ∈ V
185	183, 65, 184	fvmpt 6760	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝐺‘𝑑) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
186	185	adantl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑑) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴))))
187	180, 186	breq12d 5046	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝑑) ≤ (𝐺‘𝑑) ↔ (2↑-(!‘𝑑)) ≤ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑑 − 𝐴)))))
188	70	peano2nnd 11684	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝑑 + 1) ∈ ℕ)
189		fveq2 6659	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑑 + 1) → (!‘𝑎) = (!‘(𝑑 + 1)))
190	189	negeqd 10911	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑑 + 1) → -(!‘𝑎) = -(!‘(𝑑 + 1)))
191	190	oveq2d 7167	. . . . . . . . . 10 ⊢ (𝑎 = (𝑑 + 1) → (2↑-(!‘𝑎)) = (2↑-(!‘(𝑑 + 1))))
192		ovex 7184	. . . . . . . . . 10 ⊢ (2↑-(!‘(𝑑 + 1))) ∈ V
193	191, 5, 192	fvmpt 6760	. . . . . . . . 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 12334	. . . . . . . . . 10 ⊢ (𝑑 ∈ (ℤ_≥‘𝐴) → (𝑑 + 1) ∈ (ℤ_≥‘𝐴))
196		oveq1 7158	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑑 + 1) → (𝑐 − 𝐴) = ((𝑑 + 1) − 𝐴))
197	196	oveq2d 7167	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑑 + 1) → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑((𝑑 + 1) − 𝐴)))
198	197	oveq2d 7167	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑑 + 1) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
199		ovex 7184	. . . . . . . . . . 11 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))) ∈ V
200	198, 65, 199	fvmpt 6760	. . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → (𝐺‘(𝑑 + 1)) = ((2↑-(!‘𝐴)) · ((1 / 2)↑((𝑑 + 1) − 𝐴))))
203	194, 202	breq12d 5046	. . . . . . 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 298	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑑 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝑑) ≤ (𝐺‘𝑑) → (𝐹‘(𝑑 + 1)) ≤ (𝐺‘(𝑑 + 1))))
205	204	expcom 418	. . . . 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 12343	. . 3 ⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (𝐴 ∈ ℕ → (𝐹‘𝐵) ≤ (𝐺‘𝐵)))
208	207	impcom 412	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ≤ (𝐺‘𝐵))
209		0xr 10719	. . 3 ⊢ 0 ∈ ℝ^*
210	65	aaliou3lem1 25030	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ)
211		elioc2 12835	. . 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 591	. 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 1340	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ≠ wne 2952 class class class wbr 5033 ↦ cmpt 5113 ‘cfv 6336 (class class class)co 7151 ℂcc 10566 ℝcr 10567 0cc0 10568 1c1 10569 + caddc 10571 · cmul 10573 ℝ^*cxr 10705 < clt 10706 ≤ cle 10707 − cmin 10901 -cneg 10902 / cdiv 11328 ℕcn 11667 2c2 11722 ℕ₀cn0 11927 ℤcz 12013 ℤ_≥cuz 12275 ℝ⁺crp 12423 (,]cioc 12773 ↑cexp 13472 !cfa 13676

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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-cnex 10624 ax-resscn 10625 ax-1cn 10626 ax-icn 10627 ax-addcl 10628 ax-addrcl 10629 ax-mulcl 10630 ax-mulrcl 10631 ax-mulcom 10632 ax-addass 10633 ax-mulass 10634 ax-distr 10635 ax-i2m1 10636 ax-1ne0 10637 ax-1rid 10638 ax-rnegex 10639 ax-rrecex 10640 ax-cnre 10641 ax-pre-lttri 10642 ax-pre-lttrn 10643 ax-pre-ltadd 10644 ax-pre-mulgt0 10645

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-iun 4886 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-om 7581 df-2nd 7695 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-er 8300 df-en 8529 df-dom 8530 df-sdom 8531 df-pnf 10708 df-mnf 10709 df-xr 10710 df-ltxr 10711 df-le 10712 df-sub 10903 df-neg 10904 df-div 11329 df-nn 11668 df-2 11730 df-n0 11928 df-z 12014 df-uz 12276 df-rp 12424 df-ioc 12777 df-seq 13412 df-exp 13473 df-fac 13677

This theorem is referenced by: aaliou3lem3 25032

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