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Theorem ruclem9 16213

Description: Lemma for ruc 16218. 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 6866	. . . . . 6 ⊢ (𝑘 = 𝑀 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑀)))
3	2	breq2d 5122	. . . . 5 ⊢ (𝑘 = 𝑀 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀))))
4		2fveq3 6866	. . . . . 6 ⊢ (𝑘 = 𝑀 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑀)))
5	4	breq1d 5120	. . . . 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 6866	. . . . . 6 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
9	8	breq2d 5122	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛))))
10		2fveq3 6866	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
11	10	breq1d 5120	. . . . 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 6866	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
15	14	breq2d 5122	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
16		2fveq3 6866	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
17	16	breq1d 5120	. . . . 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 6866	. . . . . 6 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
21	20	breq2d 5122	. . . . 5 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁))))
22		2fveq3 6866	. . . . . 6 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
23	22	breq1d 5120	. . . . 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 16210	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
31		ruclem9.6	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
32	30, 31	ffvelcdmd 7060	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
33		xp1st 8003	. . . . . 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 11751	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)))
36		xp2nd 8004	. . . . . 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 11751	. . . 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 12883	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
44	31, 43	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
45	42, 44	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
46		xp1st 8003	. . . . . . . . . . 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 8004	. . . . . . . . . . 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 12488	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
51	44, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ)
52	40, 51	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
53		eqid 2730	. . . . . . . . . 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 2730	. . . . . . . . . 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 16212	. . . . . . . . . . 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 16207	. . . . . . . . 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 1142	. . . . . . . 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 16211	. . . . . . . . . . 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 8010	. . . . . . . . . . . 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 7405	. . . . . . . . . 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 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
65	64	fveq2d 6865	. . . . . . . 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 5138	. . . . . . 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 12489	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
69	44, 68	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ₀)
70	42, 69	ffvelcdmd 7060	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
71		xp1st 8003	. . . . . . . . 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 11275	. . . . . . . 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 6865	. . . . . . . 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 1144	. . . . . . . 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 5132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑛)))
79		xp2nd 8004	. . . . . . . . 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 11275	. . . . . . . 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 12876	. 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 2109 ⦋csb 3865 ∪ cun 3915 ifcif 4491 {csn 4592 ⟨cop 4598 class class class wbr 5110 × cxp 5639 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 1^st c1st 7969 2^nd c2nd 7970 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 < clt 11215 ≤ cle 11216 / cdiv 11842 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 ℤ_≥cuz 12800 seqcseq 13973

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 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 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-seq 13974

This theorem is referenced by: ruclem10 16214

W3C validator