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Theorem ruclem9 16274

Description: Lemma for ruc 16279. 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 6911	. . . . . 6 ⊢ (𝑘 = 𝑀 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑀)))
3	2	breq2d 5155	. . . . 5 ⊢ (𝑘 = 𝑀 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀))))
4		2fveq3 6911	. . . . . 6 ⊢ (𝑘 = 𝑀 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑀)))
5	4	breq1d 5153	. . . . 5 ⊢ (𝑘 = 𝑀 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
6	3, 5	anbi12d 632	. . . 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 340	. . 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 6911	. . . . . 6 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
9	8	breq2d 5155	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛))))
10		2fveq3 6911	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
11	10	breq1d 5153	. . . . 5 ⊢ (𝑘 = 𝑛 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))))
12	9, 11	anbi12d 632	. . . 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 340	. . 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 6911	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
15	14	breq2d 5155	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
16		2fveq3 6911	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
17	16	breq1d 5153	. . . . 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 632	. . . 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 340	. . 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 6911	. . . . . 6 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
21	20	breq2d 5155	. . . . 5 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁))))
22		2fveq3 6911	. . . . . 6 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
23	22	breq1d 5153	. . . . 5 ⊢ (𝑘 = 𝑁 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀))))
24	21, 23	anbi12d 632	. . . 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 340	. . 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 16271	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
31		ruclem9.6	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
32	30, 31	ffvelcdmd 7105	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
33		xp1st 8046	. . . . . 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 11829	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)))
36		xp2nd 8047	. . . . . 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 11829	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀)))
39	35, 38	jca 511	. . 3 ⊢ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)) ∧ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
40	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶ℝ)
41	27	adantr 480	. . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐺:ℕ₀⟶(ℝ × ℝ))
43		eluznn0 12959	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
44	31, 43	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
45	42, 44	ffvelcdmd 7105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
46		xp1st 8046	. . . . . . . . . . 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 8047	. . . . . . . . . . 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 12565	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
51	44, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ)
52	40, 51	ffvelcdmd 7105	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
53		eqid 2737	. . . . . . . . . 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 2737	. . . . . . . . . 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 16273	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
56	44, 55	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
57	40, 41, 47, 49, 52, 53, 54, 56	ruclem2 16268	. . . . . . . . 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 1143	. . . . . . . 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 16272	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
60	44, 59	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
61		1st2nd2 8053	. . . . . . . . . . . 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 7446	. . . . . . . . . 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 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
65	64	fveq2d 6910	. . . . . . . 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 5171	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))))
67	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
68		peano2nn0 12566	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
69	44, 68	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ₀)
70	42, 69	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
71		xp1st 8046	. . . . . . . . 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 11355	. . . . . . . 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 1373	. . . . . . 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 694	. . . . . 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 6910	. . . . . . . 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 1145	. . . . . . . 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 5165	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑛)))
79		xp2nd 8047	. . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
82		letr 11355	. . . . . . . 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 1373	. . . . . . 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 695	. . . . . 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 609	. . . . 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 413	. . . 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 12952	. 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 395 = wceq 1540 ∈ wcel 2108 ⦋csb 3899 ∪ cun 3949 ifcif 4525 {csn 4626 ⟨cop 4632 class class class wbr 5143 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1^st c1st 8012 2^nd c2nd 8013 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤ_≥cuz 12878 seqcseq 14042

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043

This theorem is referenced by: ruclem10 16275

W3C validator