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Theorem aaliou3lem7 26409

Description: Lemma for aaliou3 26411. (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 12305	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
2		eqid 2740	. . . 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 26404	. . 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 1150	. . 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 12301	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8		ax-1cn 11242	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
9		pncan 11542	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
10	7, 8, 9	sylancl 585	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((𝐴 + 1) − 1) = 𝐴)
11	10	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1...((𝐴 + 1) − 1)) = (1...𝐴))
12	11	sumeq1d 15748	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
13	12	oveq1d 7463	. . . . . . . 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 12946	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15		eqid 2740	. . . . . . . . 9 ⊢ (ℤ_≥‘(𝐴 + 1)) = (ℤ_≥‘(𝐴 + 1))
16		eqidd 2741	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) = (𝐹‘𝑏))
17		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏))
18	17	negeqd 11530	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏))
19	18	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏)))
20		ovex 7481	. . . . . . . . . . . 12 ⊢ (2↑-(!‘𝑏)) ∈ V
21	19, 3, 20	fvmpt 7029	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
22		2rp 13062	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
23		nnnn0 12560	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
24		faccl 14332	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ₀ → (!‘𝑏) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ)
26	25	nnzd 12666	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ)
27	26	znegcld 12749	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ)
28		rpexpcl 14131	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ⁺)
29	22, 27, 28	sylancr 586	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ⁺)
30	29	rpcnd 13101	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℂ)
31	21, 30	eqeltrd 2844	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℂ)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) ∈ ℂ)
33		1nn 12304	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
34		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1)))) = (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1))))
35	34, 3	aaliou3lem3 26404	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → (seq1( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘1)))))
36	35	simp1d 1142	. . . . . . . . . 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 15888	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ ℕ (𝐹‘𝑏) = (Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)))
39		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴))
40	39	sumeq1d 15748	. . . . . . . . . 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 15736	. . . . . . . . . 10 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V
43	40, 41, 42	fvmpt 7029	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
44	43	oveq1d 7463	. . . . . . . 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 2791	. . . . . . 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 2798	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = 𝐿)
48	3, 46, 41	aaliou3lem4 26406	. . . . . . . . 9 ⊢ 𝐿 ∈ ℝ
49	48	recni 11304	. . . . . . . 8 ⊢ 𝐿 ∈ ℂ
50	49	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐿 ∈ ℂ)
51	3, 46, 41	aaliou3lem5 26407	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ)
52	51	recnd 11318	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℂ)
53	4	simp2d 1143	. . . . . . . . 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 13101	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℂ)
56	50, 52, 55	subaddd 11665	. . . . . 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 257	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐿 − (𝐻‘𝐴)) = Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏))
58	57	eqcomd 2746	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)))
59		eleq1 2832	. . . . 5 ⊢ (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)) → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
60		breq1 5169	. . . . 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 631	. . . 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 480	. . . . . 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 13095	. . . . . . . 8 ⊢ (((𝐻‘𝐴) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝐻‘𝐴) < 𝐿 ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
66	63, 48, 65	sylancl 585	. . . . . . 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 257	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) < 𝐿)
68	63, 67	ltned 11426	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ≠ 𝐿)
69		nnnn0 12560	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ → (𝐴 + 1) ∈ ℕ₀)
70		faccl 14332	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ₀ → (!‘(𝐴 + 1)) ∈ ℕ)
71	1, 69, 70	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℕ)
72	71	nnzd 12666	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℤ)
73	72	znegcld 12749	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → -(!‘(𝐴 + 1)) ∈ ℤ)
74		rpexpcl 14131	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘(𝐴 + 1)) ∈ ℤ) → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
75	22, 73, 74	sylancr 586	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
76		rpmulcl 13080	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ⁺ ∧ (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
77	22, 75, 76	sylancr 586	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
78	77	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
79	78	rpred 13099	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ)
80	63, 79	resubcld 11718	. . . . . . 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 13131	. . . . . . . 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 11451	. . . . . . 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 11438	. . . . . 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 11889	. . . . . . 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 232	. . . . . 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 15483	. . . . . 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 511	. . . 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 412	. . 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 240	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 ...cfz 13567 seqcseq 14052 ↑cexp 14112 !cfa 14322 abscabs 15283 ⇝ cli 15530 Σcsu 15734

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-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 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 ax-pre-sup 11262

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-int 4971 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-se 5653 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-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 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-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ioc 13412 df-ico 13413 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-fac 14323 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735

This theorem is referenced by: aaliou3lem9 26410

W3C validator