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Theorem ruclem9 16272

Description: Lemma for ruc 16277. 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 6874	. . . . . 6 ⊢ (𝑘 = 𝑀 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑀)))
3	2	breq2d 5114	. . . . 5 ⊢ (𝑘 = 𝑀 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀))))
4		2fveq3 6874	. . . . . 6 ⊢ (𝑘 = 𝑀 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑀)))
5	4	breq1d 5112	. . . . 5 ⊢ (𝑘 = 𝑀 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
6	3, 5	anbi12d 641	. . . 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 342	. . 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 6874	. . . . . 6 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
9	8	breq2d 5114	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛))))
10		2fveq3 6874	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
11	10	breq1d 5112	. . . . 5 ⊢ (𝑘 = 𝑛 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))))
12	9, 11	anbi12d 641	. . . 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 342	. . 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 6874	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
15	14	breq2d 5114	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
16		2fveq3 6874	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
17	16	breq1d 5112	. . . . 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 641	. . . 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 342	. . 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 6874	. . . . . 6 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
21	20	breq2d 5114	. . . . 5 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁))))
22		2fveq3 6874	. . . . . 6 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
23	22	breq1d 5112	. . . . 5 ⊢ (𝑘 = 𝑁 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀))))
24	21, 23	anbi12d 641	. . . 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 342	. . 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 16269	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
31		ruclem9.6	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
32	30, 31	ffvelcdmd 7068	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
33		xp1st 8004	. . . . . 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 11755	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)))
36		xp2nd 8005	. . . . . 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 11755	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀)))
39	35, 38	jca 519	. . 3 ⊢ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)) ∧ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
40	26	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶ℝ)
41	27	adantr 484	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐺:ℕ₀⟶(ℝ × ℝ))
43		eluznn0 12920	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
44	31, 43	sylan 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
45	42, 44	ffvelcdmd 7068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
46		xp1st 8004	. . . . . . . . . . 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 8005	. . . . . . . . . . 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 12522	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
51	44, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ)
52	40, 51	ffvelcdmd 7068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
53		eqid 2764	. . . . . . . . . 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 2764	. . . . . . . . . 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 16271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
56	44, 55	syldan 600	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
57	40, 41, 47, 49, 52, 53, 54, 56	ruclem2 16266	. . . . . . . . 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 1156	. . . . . . . 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 16270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
60	44, 59	syldan 600	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
61		1st2nd2 8011	. . . . . . . . . . . 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 7413	. . . . . . . . . 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 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
65	64	fveq2d 6873	. . . . . . . 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 5130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))))
67	34	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
68		peano2nn0 12523	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
69	44, 68	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ₀)
70	42, 69	ffvelcdmd 7068	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
71		xp1st 8004	. . . . . . . . 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 11279	. . . . . . . 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 1392	. . . . . . 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 704	. . . . . 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 6873	. . . . . . . 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 1158	. . . . . . . 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 5124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑛)))
79		xp2nd 8005	. . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
82		letr 11279	. . . . . . . 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 1392	. . . . . . 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 705	. . . . . 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 618	. . . . 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 417	. . . 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 12913	. 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 399 = wceq 1562 ∈ wcel 2144 ⦋csb 3854 ∪ cun 3904 ifcif 4482 {csn 4584 ⟨cop 4590 class class class wbr 5102 × cxp 5647 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ∈ cmpo 7400 1^st c1st 7970 2^nd c2nd 7971 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 < clt 11218 ≤ cle 11219 / cdiv 11846 ℕcn 12212 2c2 12274 ℕ₀cn0 12483 ℤ_≥cuz 12841 seqcseq 14016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-n0 12484 df-z 12571 df-uz 12842 df-fz 13515 df-seq 14017

This theorem is referenced by: ruclem10 16273

W3C validator