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Theorem aaliou3lem7 26586

Description: Lemma for aaliou3 26588. (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 12270	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
2		eqid 2760	. . . 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 26581	. . 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 1168	. . 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 19	. 2 ⊢ (𝐴 ∈ ℕ → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))
7		nncn 12266	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8		ax-1cn 11183	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
9		pncan 11488	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
10	7, 8, 9	sylancl 598	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((𝐴 + 1) − 1) = 𝐴)
11	10	oveq2d 7430	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1...((𝐴 + 1) − 1)) = (1...𝐴))
12	11	sumeq1d 15788	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
13	12	oveq1d 7429	. . . . . . . 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 12927	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15		eqid 2760	. . . . . . . . 9 ⊢ (ℤ_≥‘(𝐴 + 1)) = (ℤ_≥‘(𝐴 + 1))
16		eqidd 2761	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) = (𝐹‘𝑏))
17		fveq2 6879	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏))
18	17	negeqd 11476	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏))
19	18	oveq2d 7430	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏)))
20		ovex 7447	. . . . . . . . . . . 12 ⊢ (2↑-(!‘𝑏)) ∈ V
21	19, 3, 20	fvmpt 6987	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
22		2rp 13048	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
23		nnnn0 12536	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
24		faccl 14348	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ₀ → (!‘𝑏) ∈ ℕ)
25	23, 24	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ)
26	25	nnzd 12642	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ)
27	26	znegcld 12728	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ)
28		rpexpcl 14145	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ⁺)
29	22, 27, 28	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ⁺)
30	29	rpcnd 13089	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℂ)
31	21, 30	eqeltrd 2860	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℂ)
32	31	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) ∈ ℂ)
33		1nn 12269	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
34		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1)))) = (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1))))
35	34, 3	aaliou3lem3 26581	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → (seq1( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘1)))))
36	35	simp1d 1160	. . . . . . . . . 10 ⊢ (1 ∈ ℕ → seq1( + , 𝐹) ∈ dom ⇝ )
37	33, 36	mp1i 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → seq1( + , 𝐹) ∈ dom ⇝ )
38	14, 15, 1, 16, 32, 37	isumsplit 15930	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ ℕ (𝐹‘𝑏) = (Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)))
39		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴))
40	39	sumeq1d 15788	. . . . . . . . . 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 15776	. . . . . . . . . 10 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V
43	40, 41, 42	fvmpt 6987	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
44	43	oveq1d 7429	. . . . . . . 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 2806	. . . . . . 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 2813	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = 𝐿)
48	3, 46, 41	aaliou3lem4 26583	. . . . . . . . 9 ⊢ 𝐿 ∈ ℝ
49	48	recni 11248	. . . . . . . 8 ⊢ 𝐿 ∈ ℂ
50	49	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐿 ∈ ℂ)
51	3, 46, 41	aaliou3lem5 26584	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ)
52	51	recnd 11262	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℂ)
53	4	simp2d 1161	. . . . . . . . 9 ⊢ ((𝐴 + 1) ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺)
54	1, 53	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺)
55	54	rpcnd 13089	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℂ)
56	50, 52, 55	subaddd 11612	. . . . . 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 2766	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)))
59		eleq1 2848	. . . . 5 ⊢ (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)) → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
60		breq1 5106	. . . . 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 644	. . . 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 18	. . 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 486	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ∈ ℝ)
64		simprl 783	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺)
65		difrp 13083	. . . . . . . 8 ⊢ (((𝐻‘𝐴) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝐻‘𝐴) < 𝐿 ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
66	63, 48, 65	sylancl 598	. . . . . . 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 11371	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ≠ 𝐿)
69		nnnn0 12536	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ → (𝐴 + 1) ∈ ℕ₀)
70		faccl 14348	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ₀ → (!‘(𝐴 + 1)) ∈ ℕ)
71	1, 69, 70	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℕ)
72	71	nnzd 12642	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℤ)
73	72	znegcld 12728	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → -(!‘(𝐴 + 1)) ∈ ℤ)
74		rpexpcl 14145	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘(𝐴 + 1)) ∈ ℤ) → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
75	22, 73, 74	sylancr 599	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
76		rpmulcl 13068	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ⁺ ∧ (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
77	22, 75, 76	sylancr 599	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
78	77	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
79	78	rpred 13087	. . . . . . . 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 13119	. . . . . . . 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 11396	. . . . . . 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 11383	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) ≤ 𝐿)
85		simprr 785	. . . . . . 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 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 15525	. . . . . 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 726	. . . . 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 521	. . . 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 418	. . 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 16	1 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻‘𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 1c1 11126 + caddc 11128 · cmul 11130 < clt 11268 ≤ cle 11269 − cmin 11466 -cneg 11467 / cdiv 11896 ℕcn 12258 2c2 12320 ℕ₀cn0 12529 ℤcz 12616 ℤ_≥cuz 12888 ℝ⁺crp 13043 ...cfz 13562 seqcseq 14066 ↑cexp 14126 !cfa 14338 abscabs 15322 ⇝ cli 15572 Σcsu 15774

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-pm 8830 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9413 df-inf 9414 df-oi 9483 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-z 12617 df-uz 12889 df-rp 13044 df-ioc 13404 df-ico 13405 df-fz 13563 df-fzo 13711 df-fl 13854 df-seq 14067 df-exp 14127 df-fac 14339 df-hash 14396 df-shft 15141 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-limsup 15559 df-clim 15576 df-rlim 15577 df-sum 15775

This theorem is used by: aaliou3lem9 26587

W3C validator