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Theorem aaliou3lem7 26406

Description: Lemma for aaliou3 26408. (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 12276	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
2		eqid 2735	. . . 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 26401	. . 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 1149	. . 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 12272	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8		ax-1cn 11211	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
9		pncan 11512	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
10	7, 8, 9	sylancl 586	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((𝐴 + 1) − 1) = 𝐴)
11	10	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1...((𝐴 + 1) − 1)) = (1...𝐴))
12	11	sumeq1d 15733	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
13	12	oveq1d 7446	. . . . . . . 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 12919	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15		eqid 2735	. . . . . . . . 9 ⊢ (ℤ_≥‘(𝐴 + 1)) = (ℤ_≥‘(𝐴 + 1))
16		eqidd 2736	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) = (𝐹‘𝑏))
17		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏))
18	17	negeqd 11500	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏))
19	18	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏)))
20		ovex 7464	. . . . . . . . . . . 12 ⊢ (2↑-(!‘𝑏)) ∈ V
21	19, 3, 20	fvmpt 7016	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
22		2rp 13037	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
23		nnnn0 12531	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
24		faccl 14319	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ℕ₀ → (!‘𝑏) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ)
26	25	nnzd 12638	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ)
27	26	znegcld 12722	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ)
28		rpexpcl 14118	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ⁺)
29	22, 27, 28	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ⁺)
30	29	rpcnd 13077	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℂ)
31	21, 30	eqeltrd 2839	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℂ)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝐹‘𝑏) ∈ ℂ)
33		1nn 12275	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
34		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1)))) = (𝑐 ∈ (ℤ_≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1))))
35	34, 3	aaliou3lem3 26401	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → (seq1( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘1)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘1)))))
36	35	simp1d 1141	. . . . . . . . . 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 15873	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ ℕ (𝐹‘𝑏) = (Σ𝑏 ∈ (1...((𝐴 + 1) − 1))(𝐹‘𝑏) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)))
39		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴))
40	39	sumeq1d 15733	. . . . . . . . . 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 15721	. . . . . . . . . 10 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V
43	40, 41, 42	fvmpt 7016	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
44	43	oveq1d 7446	. . . . . . . 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 2786	. . . . . . 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 2793	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) + Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏)) = 𝐿)
48	3, 46, 41	aaliou3lem4 26403	. . . . . . . . 9 ⊢ 𝐿 ∈ ℝ
49	48	recni 11273	. . . . . . . 8 ⊢ 𝐿 ∈ ℂ
50	49	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐿 ∈ ℂ)
51	3, 46, 41	aaliou3lem5 26404	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ)
52	51	recnd 11287	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℂ)
53	4	simp2d 1142	. . . . . . . . 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 13077	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℂ)
56	50, 52, 55	subaddd 11636	. . . . . 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 2741	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)))
59		eleq1 2827	. . . . 5 ⊢ (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) = (𝐿 − (𝐻‘𝐴)) → (Σ𝑏 ∈ (ℤ_≥‘(𝐴 + 1))(𝐹‘𝑏) ∈ ℝ⁺ ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
60		breq1 5151	. . . . 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 632	. . . 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 771	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺)
65		difrp 13071	. . . . . . . 8 ⊢ (((𝐻‘𝐴) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝐻‘𝐴) < 𝐿 ↔ (𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺))
66	63, 48, 65	sylancl 586	. . . . . . 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 11395	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (𝐻‘𝐴) ≠ 𝐿)
69		nnnn0 12531	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ → (𝐴 + 1) ∈ ℕ₀)
70		faccl 14319	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + 1) ∈ ℕ₀ → (!‘(𝐴 + 1)) ∈ ℕ)
71	1, 69, 70	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℕ)
72	71	nnzd 12638	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (!‘(𝐴 + 1)) ∈ ℤ)
73	72	znegcld 12722	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → -(!‘(𝐴 + 1)) ∈ ℤ)
74		rpexpcl 14118	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘(𝐴 + 1)) ∈ ℤ) → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
75	22, 73, 74	sylancr 587	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺)
76		rpmulcl 13056	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ⁺ ∧ (2↑-(!‘(𝐴 + 1))) ∈ ℝ⁺) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ⁺)
77	22, 75, 76	sylancr 587	. . . . . . . . . 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 13075	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → (2 · (2↑-(!‘(𝐴 + 1)))) ∈ ℝ)
80	63, 79	resubcld 11689	. . . . . . 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 13107	. . . . . . . 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 11420	. . . . . . 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 11407	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ ((𝐿 − (𝐻‘𝐴)) ∈ ℝ⁺ ∧ (𝐿 − (𝐻‘𝐴)) ≤ (2 · (2↑-(!‘(𝐴 + 1)))))) → ((𝐻‘𝐴) − (2 · (2↑-(!‘(𝐴 + 1))))) ≤ 𝐿)
85		simprr 773	. . . . . . 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 11860	. . . . . . 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 15470	. . . . . 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 713	. . . . 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 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ℝ⁺crp 13032 ...cfz 13544 seqcseq 14039 ↑cexp 14099 !cfa 14309 abscabs 15270 ⇝ cli 15517 Σcsu 15719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ioc 13389 df-ico 13390 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-fac 14310 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720

This theorem is referenced by: aaliou3lem9 26407

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