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Theorem aaliou3lem7 24945

Description: Lemma for aaliou3 24947. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
aaliou3lem.c	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
aaliou3lem.d	⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏)
aaliou3lem.e	⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏))

**Assertion**
Ref	Expression
aaliou3lem7	⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))

Distinct variable groups: 𝑎,𝑏,𝑐 𝐹,𝑏,𝑐 𝐿,𝑐 𝐴,𝑎,𝑏,𝑐 Allowed substitution hints: 𝐹(𝑎) 𝐻(𝑎,𝑏,𝑐) 𝐿(𝑎,𝑏)

**Proof of Theorem aaliou3lem7**
Step	Hyp	Ref	Expression
1		peano2nn 11637	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
2		eqid 2798	. . . 4 ⊢ (𝑐 ∈ (ℤ_≥‘(𝐴 + 1)) ↦ ((2↑-(!‘(𝐴 + 1))) · ((1 / 2)↑(𝑐 − (𝐴 + 1))))) = (𝑐 ∈ (ℤ_≥‘(𝐴 + 1)) ↦ ((2↑-(!‘(𝐴 + 1))) · ((1 / 2)↑(𝑐 − (𝐴 + 1)))))
3		aaliou3lem.c	. . . 4 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
4	2, 3	aaliou3lem3 24940	. . 3 ⊢ ((𝐴 + 1) ∈ ℕ → (seq(𝐴 + 1)( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))
5		3simpc 1147	. . 3 ⊢ ((seq(𝐴 + 1)( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))) → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))
6	1, 4, 5	3syl 18	. 2 ⊢ (𝐴 ∈ ℕ → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))
7		nncn 11633	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8		ax-1cn 10584	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
9		pncan 10881	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
10	7, 8, 9	sylancl 589	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((𝐴 + 1) − 1) = 𝐴)
11	10	oveq2d 7151	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1...((𝐴 + 1) − 1)) = (1...𝐴))
12	11	sumeq1d 15050	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
13	12	oveq1d 7150	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = (Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)))
14		nnuz 12269	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15		eqid 2798	. . . . . . . . 9 ⊢ (ℤ_≥‘(𝐴 + 1)) = (ℤ_≥‘(𝐴 + 1))
16		eqidd 2799	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) = (𝐹‘𝑏))
17		fveq2 6645	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏))
18	17	negeqd 10869	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏))
19	18	oveq2d 7151	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏)))
20		ovex 7168	. . . . . . . . . . . 12 ⊢ (2↑-(!‘𝑏)) ∈ V
21	19, 3, 20	fvmpt 6745	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
22		2rp 12382	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
23		nnnn0 11892	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
24		faccl 13639	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ₀ → (!‘𝑏) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ)
26	25	nnzd 12074	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ)
27	26	znegcld 12077	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ)
28		rpexpcl 13444	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ⁺)
29	22, 27, 28	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ⁺)
30	29	rpcnd 12421	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℂ)
31	21, 30	eqeltrd 2890	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℂ)
32	31	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) ∈ ℂ)
33		1nn 11636	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
34		eqid 2798	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1)))) = (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1))))
35	34, 3	aaliou3lem3 24940	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → (seq1( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘1)))))
36	35	simp1d 1139	. . . . . . . . . 10 ⊢ (1 ∈ ℕ → seq1( + , 𝐹) ∈ dom ⇝ )
37	33, 36	mp1i 13	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → seq1( + , 𝐹) ∈ dom ⇝ )
38	14, 15, 1, 16, 32, 37	isumsplit 15187	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ ℕ (𝐹‘𝑏) = (Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)))
39		oveq2 7143	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴))
40	39	sumeq1d 15050	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
41		aaliou3lem.e	. . . . . . . . . 10 ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏))
42		sumex 15036	. . . . . . . . . 10 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V
43	40, 41, 42	fvmpt 6745	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
44	43	oveq1d 7150	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = (Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)))
45	13, 38, 44	3eqtr4rd 2844	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = Σ𝑏 ∈ ℕ (𝐹‘𝑏))
46		aaliou3lem.d	. . . . . . 7 ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏)
47	45, 46	eqtr4di 2851	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = 𝐿)
48	3, 46, 41	aaliou3lem4 24942	. . . . . . . . 9 ⊢ 𝐿 ∈ ℝ
49	48	recni 10644	. . . . . . . 8 ⊢ 𝐿 ∈ ℂ
50	49	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐿 ∈ ℂ)
51	3, 46, 41	aaliou3lem5 24943	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ)
52	51	recnd 10658	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℂ)
53	4	simp2d 1140	. . . . . . . . 9 ⊢ ((𝐴 + 1) ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺)
54	1, 53	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺)
55	54	rpcnd 12421	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℂ)
56	50, 52, 55	subaddd 11004	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐿 − (𝐻‘𝐴)) = Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ↔ ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = 𝐿))
57	47, 56	mpbird 260	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐿 − (𝐻‘𝐴)) = Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏))
58	57	eqcomd 2804	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)))
59		eleq1 2877	. . . . 5 ⊢ (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)) → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
60		breq1 5033	. . . . 5 ⊢ (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)) → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1)))) ↔ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))
61	59, 60	anbi12d 633	. . . 4 ⊢ (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)) → ((Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))) ↔ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))))
62	58, 61	syl 17	. . 3 ⊢ (𝐴 ∈ ℕ → ((Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))) ↔ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))))
63	51	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ∈ ℝ)
64		simprl 770	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺)
65		difrp 12415	. . . . . . . 8 ⊢ (((𝐻‘𝐴) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝐻‘𝐴) < 𝐿 ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
66	63, 48, 65	sylancl 589	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) < 𝐿 ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
67	64, 66	mpbird 260	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) < 𝐿)
68	63, 67	ltned 10765	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ≠ 𝐿)
69		nnnn0 11892	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ → (𝐴 + 1) ∈ ℕ₀)
70		faccl 13639	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ₀ → (!‘(𝐴 + 1)) ∈ ℕ)
71	1, 69, 70	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℕ)
72	71	nnzd 12074	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℤ)
73	72	znegcld 12077	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → -(!‘(𝐴 + 1)) ∈ ℤ)
74		rpexpcl 13444	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘(𝐴 + 1)) ∈ ℤ) → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
75	22, 73, 74	sylancr 590	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
76		rpmulcl 12400	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ⁺ ∧ (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
77	22, 75, 76	sylancr 590	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
78	77	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
79	78	rpred 12419	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ)
80	63, 79	resubcld 11057	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) ∈ ℝ)
81	48	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → 𝐿 ∈ ℝ)
82	63, 78	ltsubrpd 12451	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) < (𝐻‘𝐴))
83	80, 63, 81, 82, 67	lttrd 10790	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) < 𝐿)
84	80, 81, 83	ltled 10777	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) ≤ 𝐿)
85		simprr 772	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))
86	81, 63, 79	lesubadd2d 11228	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))) ↔ 𝐿 ≤ ((𝐻‘𝐴) + (2 · (2↑-(!‘(𝐴 + 1)))))))
87	85, 86	mpbid 235	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → 𝐿 ≤ ((𝐻‘𝐴) + (2 · (2↑-(!‘(𝐴 + 1))))))
88	81, 63, 79	absdifled 14786	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1)))) ↔ (((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) ≤ 𝐿 ∧ 𝐿 ≤ ((𝐻‘𝐴) + (2 · (2↑-(!‘(𝐴 + 1))))))))
89	84, 87, 88	mpbir2and 712	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))
90	68, 89	jca 515	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))
91	90	ex 416	. . 3 ⊢ (𝐴 ∈ ℕ → (((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1))))) → ((𝐻‘𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))))
92	62, 91	sylbid 243	. 2 ⊢ (𝐴 ∈ ℕ → ((Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))) → ((𝐻‘𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))))
93	6, 92	mpd 15	1 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 ...cfz 12885 seqcseq 13364 ↑cexp 13425 !cfa 13629 abscabs 14585 ⇝ cli 14833 Σcsu 15034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ioc 12731 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035

This theorem is referenced by: aaliou3lem9 24946

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