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Theorem aaliou3lem7 26297

Description: Lemma for aaliou3 26299. (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 12249	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
2		eqid 2725	. . . 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 26292	. . 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 12245	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8		ax-1cn 11191	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
9		pncan 11491	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
10	7, 8, 9	sylancl 584	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((𝐴 + 1) − 1) = 𝐴)
11	10	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1...((𝐴 + 1) − 1)) = (1...𝐴))
12	11	sumeq1d 15674	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
13	12	oveq1d 7428	. . . . . . . 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 12890	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15		eqid 2725	. . . . . . . . 9 ⊢ (ℤ_≥‘(𝐴 + 1)) = (ℤ_≥‘(𝐴 + 1))
16		eqidd 2726	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) = (𝐹‘𝑏))
17		fveq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏))
18	17	negeqd 11479	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏))
19	18	oveq2d 7429	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏)))
20		ovex 7446	. . . . . . . . . . . 12 ⊢ (2↑-(!‘𝑏)) ∈ V
21	19, 3, 20	fvmpt 6998	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
22		2rp 13006	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
23		nnnn0 12504	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
24		faccl 14269	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ₀ → (!‘𝑏) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ)
26	25	nnzd 12610	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ)
27	26	znegcld 12693	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ)
28		rpexpcl 14072	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ⁺)
29	22, 27, 28	sylancr 585	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ⁺)
30	29	rpcnd 13045	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℂ)
31	21, 30	eqeltrd 2825	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℂ)
32	31	adantl 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) ∈ ℂ)
33		1nn 12248	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
34		eqid 2725	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1)))) = (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1))))
35	34, 3	aaliou3lem3 26292	. . . . . . . . . . 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 15813	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ ℕ (𝐹‘𝑏) = (Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)))
39		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴))
40	39	sumeq1d 15674	. . . . . . . . . 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 15661	. . . . . . . . . 10 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V
43	40, 41, 42	fvmpt 6998	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
44	43	oveq1d 7428	. . . . . . . 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 2776	. . . . . . 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 2783	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = 𝐿)
48	3, 46, 41	aaliou3lem4 26294	. . . . . . . . 9 ⊢ 𝐿 ∈ ℝ
49	48	recni 11253	. . . . . . . 8 ⊢ 𝐿 ∈ ℂ
50	49	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐿 ∈ ℂ)
51	3, 46, 41	aaliou3lem5 26295	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ)
52	51	recnd 11267	. . . . . . 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 13045	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℂ)
56	50, 52, 55	subaddd 11614	. . . . . 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 256	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐿 − (𝐻‘𝐴)) = Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏))
58	57	eqcomd 2731	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)))
59		eleq1 2813	. . . . 5 ⊢ (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)) → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
60		breq1 5147	. . . . 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 630	. . . 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 479	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ∈ ℝ)
64		simprl 769	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺)
65		difrp 13039	. . . . . . . 8 ⊢ (((𝐻‘𝐴) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝐻‘𝐴) < 𝐿 ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
66	63, 48, 65	sylancl 584	. . . . . . 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 256	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) < 𝐿)
68	63, 67	ltned 11375	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ≠ 𝐿)
69		nnnn0 12504	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ → (𝐴 + 1) ∈ ℕ₀)
70		faccl 14269	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ₀ → (!‘(𝐴 + 1)) ∈ ℕ)
71	1, 69, 70	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℕ)
72	71	nnzd 12610	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℤ)
73	72	znegcld 12693	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → -(!‘(𝐴 + 1)) ∈ ℤ)
74		rpexpcl 14072	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘(𝐴 + 1)) ∈ ℤ) → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
75	22, 73, 74	sylancr 585	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
76		rpmulcl 13024	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ⁺ ∧ (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
77	22, 75, 76	sylancr 585	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
78	77	adantr 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
79	78	rpred 13043	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ)
80	63, 79	resubcld 11667	. . . . . . 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 13075	. . . . . . . 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 11400	. . . . . . 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 11387	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) ≤ 𝐿)
85		simprr 771	. . . . . . 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 11838	. . . . . . 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 231	. . . . . 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 15408	. . . . . 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 711	. . . . 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 510	. . . 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 411	. . 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 239	. 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5144 ↦ cmpt 5227 dom cdm 5673 ‘cfv 6543 (class class class)co 7413 ℂcc 11131 ℝcr 11132 1c1 11134 + caddc 11136 · cmul 11138 < clt 11273 ≤ cle 11274 − cmin 11469 -cneg 11470 / cdiv 11896 ℕcn 12237 2c2 12292 ℕ₀cn0 12497 ℤcz 12583 ℤ_≥cuz 12847 ℝ⁺crp 13001 ...cfz 13511 seqcseq 13993 ↑cexp 14053 !cfa 14259 abscabs 15208 ⇝ cli 15455 Σcsu 15659

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-ioc 13356 df-ico 13357 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-fac 14260 df-hash 14317 df-shft 15041 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660

This theorem is referenced by: aaliou3lem9 26298

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