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Theorem ruclem9 16235

Description: Lemma for ruc 16240. The first components of the 𝐺 sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)

**Hypotheses**
Ref	Expression
ruc.1	⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
ruc.2	⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
ruc.4	⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5	⊢ 𝐺 = seq0(𝐷, 𝐶)
ruclem9.6	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
ruclem9.7	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))

**Assertion**
Ref	Expression
ruclem9	⊢ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁)) ∧ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀))))

Distinct variable groups: 𝑥,𝑚,𝑦,𝐹 𝑚,𝐺,𝑥,𝑦 𝑚,𝑀,𝑥,𝑦 𝑚,𝑁,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑚) 𝐶(𝑥,𝑦,𝑚) 𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem9 Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ruclem9.7	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = 𝑀 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑀)))
3	2	breq2d 5157	. . . . 5 ⊢ (𝑘 = 𝑀 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀))))
4		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = 𝑀 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑀)))
5	4	breq1d 5155	. . . . 5 ⊢ (𝑘 = 𝑀 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
6	3, 5	anbi12d 630	. . . 4 ⊢ (𝑘 = 𝑀 → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀))) ↔ ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)) ∧ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀)))))
7	6	imbi2d 339	. . 3 ⊢ (𝑘 = 𝑀 → ((𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)) ∧ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))))
8		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
9	8	breq2d 5157	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛))))
10		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
11	10	breq1d 5155	. . . . 5 ⊢ (𝑘 = 𝑛 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))))
12	9, 11	anbi12d 630	. . . 4 ⊢ (𝑘 = 𝑛 → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀))) ↔ ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀)))))
13	12	imbi2d 339	. . 3 ⊢ (𝑘 = 𝑛 → ((𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))))))
14		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
15	14	breq2d 5157	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
16		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
17	16	breq1d 5155	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))
18	15, 17	anbi12d 630	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀))) ↔ ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))) ∧ (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀)))))
19	18	imbi2d 339	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))) ∧ (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))))
20		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
21	20	breq2d 5157	. . . . 5 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁))))
22		2fveq3 6898	. . . . . 6 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
23	22	breq1d 5155	. . . . 5 ⊢ (𝑘 = 𝑁 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀))))
24	21, 23	anbi12d 630	. . . 4 ⊢ (𝑘 = 𝑁 → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀))) ↔ ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁)) ∧ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀)))))
25	24	imbi2d 339	. . 3 ⊢ (𝑘 = 𝑁 → ((𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ∧ (2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁)) ∧ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀))))))
26		ruc.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
27		ruc.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
28		ruc.4	. . . . . . . 8 ⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
29		ruc.5	. . . . . . . 8 ⊢ 𝐺 = seq0(𝐷, 𝐶)
30	26, 27, 28, 29	ruclem6 16232	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
31		ruclem9.6	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
32	30, 31	ffvelcdmd 7091	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
33		xp1st 8027	. . . . . 6 ⊢ ((𝐺‘𝑀) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
34	32, 33	syl 17	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
35	34	leidd 11821	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)))
36		xp2nd 8028	. . . . . 6 ⊢ ((𝐺‘𝑀) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
37	32, 36	syl 17	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
38	37	leidd 11821	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀)))
39	35, 38	jca 510	. . 3 ⊢ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)) ∧ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
40	26	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶ℝ)
41	27	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
42	30	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐺:ℕ₀⟶(ℝ × ℝ))
43		eluznn0 12947	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
44	31, 43	sylan 578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
45	42, 44	ffvelcdmd 7091	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
46		xp1st 8027	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
48		xp2nd 8028	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
49	45, 48	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
50		nn0p1nn 12557	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
51	44, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ)
52	40, 51	ffvelcdmd 7091	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
53		eqid 2726	. . . . . . . . . 10 ⊢ (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
54		eqid 2726	. . . . . . . . . 10 ⊢ (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
55	26, 27, 28, 29	ruclem8 16234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
56	44, 55	syldan 589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
57	40, 41, 47, 49, 52, 53, 54, 56	ruclem2 16229	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ∧ (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) < (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ∧ (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2^nd ‘(𝐺‘𝑛))))
58	57	simp1d 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
59	26, 27, 28, 29	ruclem7 16233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
60	44, 59	syldan 589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
61		1st2nd2 8034	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩)
62	45, 61	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩)
63	62	oveq1d 7431	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
64	60, 63	eqtrd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
65	64	fveq2d 6897	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘(𝑛 + 1))) = (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
66	58, 65	breqtrrd 5173	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))))
67	34	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
68		peano2nn0 12558	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
69	44, 68	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ₀)
70	42, 69	ffvelcdmd 7091	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
71		xp1st 8027	. . . . . . . . 9 ⊢ ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
72	70, 71	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
73		letr 11349	. . . . . . . 8 ⊢ (((1^st ‘(𝐺‘𝑀)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ) → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) ∧ (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))) → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
74	67, 47, 72, 73	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) ∧ (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))) → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
75	66, 74	mpan2d 692	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
76	64	fveq2d 6897	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) = (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
77	57	simp3d 1141	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2^nd ‘(𝐺‘𝑛)))
78	76, 77	eqbrtrd 5167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑛)))
79		xp2nd 8028	. . . . . . . . 9 ⊢ ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
80	70, 79	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
81	37	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
82		letr 11349	. . . . . . . 8 ⊢ (((2^nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝑀)) ∈ ℝ) → (((2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))
83	80, 49, 81, 82	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (((2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))
84	78, 83	mpand 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))
85	75, 84	anim12d 607	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))) ∧ (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀)))))
86	85	expcom 412	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝜑 → (((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))) ∧ (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))))
87	86	a2d 29	. . 3 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀)))) → (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))) ∧ (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))))
88	7, 13, 19, 25, 39, 87	uzind4i 12940	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁)) ∧ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀)))))
89	1, 88	mpcom 38	1 ⊢ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁)) ∧ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ⦋csb 3891 ∪ cun 3944 ifcif 4523 {csn 4623 ⟨cop 4629 class class class wbr 5145 × cxp 5672 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ∈ cmpo 7418 1^st c1st 7993 2^nd c2nd 7994 ℝcr 11148 0cc0 11149 1c1 11150 + caddc 11152 < clt 11289 ≤ cle 11290 / cdiv 11912 ℕcn 12258 2c2 12313 ℕ₀cn0 12518 ℤ_≥cuz 12868 seqcseq 14015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-seq 14016

This theorem is referenced by: ruclem10 16236

W3C validator