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Theorem ruclem9 14806

Description: Lemma for ruc 14811. 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		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀))
3	2	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 𝑀 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑀)))
4	3	breq2d 4595	. . . . 5 ⊢ (𝑘 = 𝑀 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀))))
5	2	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 𝑀 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑀)))
6	5	breq1d 4593	. . . . 5 ⊢ (𝑘 = 𝑀 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
7	4, 6	anbi12d 743	. . . 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)))))
8	7	imbi2d 329	. . 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))))))
9		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐺‘𝑘) = (𝐺‘𝑛))
10	9	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
11	10	breq2d 4595	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛))))
12	9	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
13	12	breq1d 4593	. . . . 5 ⊢ (𝑘 = 𝑛 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀))))
14	11, 13	anbi12d 743	. . . 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)))))
15	14	imbi2d 329	. . 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))))))
16		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
17	16	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
18	17	breq2d 4595	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
19	16	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
20	19	breq1d 4593	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))
21	18, 20	anbi12d 743	. . . 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)))))
22	21	imbi2d 329	. . 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))))))
23		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (𝐺‘𝑘) = (𝐺‘𝑁))
24	23	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
25	24	breq2d 4595	. . . . 5 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁))))
26	23	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
27	26	breq1d 4593	. . . . 5 ⊢ (𝑘 = 𝑁 → ((2^nd ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑀)) ↔ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀))))
28	25, 27	anbi12d 743	. . . 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)))))
29	28	imbi2d 329	. . 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))))))
30		ruc.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
31		ruc.2	. . . . . . . . 9 ⊢ (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)⟩)))
32		ruc.4	. . . . . . . . 9 ⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
33		ruc.5	. . . . . . . . 9 ⊢ 𝐺 = seq0(𝐷, 𝐶)
34	30, 31, 32, 33	ruclem6 14803	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
35		ruclem9.6	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
36	34, 35	ffvelrnd 6268	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
37		xp1st 7089	. . . . . . 7 ⊢ ((𝐺‘𝑀) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
39	38	leidd 10473	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)))
40		xp2nd 7090	. . . . . . 7 ⊢ ((𝐺‘𝑀) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
41	36, 40	syl 17	. . . . . 6 ⊢ (𝜑 → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
42	41	leidd 10473	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀)))
43	39, 42	jca 553	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)) ∧ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀))))
44	43	a1i 11	. . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑀)) ∧ (2^nd ‘(𝐺‘𝑀)) ≤ (2^nd ‘(𝐺‘𝑀)))))
45	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶ℝ)
46	31	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)⟩)))
47	34	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝐺:ℕ₀⟶(ℝ × ℝ))
48		eluznn0 11633	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
49	35, 48	sylan 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℕ₀)
50	47, 49	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
51		xp1st 7089	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
53		xp2nd 7090	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
54	50, 53	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
55		nn0p1nn 11209	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
56	49, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ)
57	45, 56	ffvelrnd 6268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
58		eqid 2610	. . . . . . . . . 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))))
59		eqid 2610	. . . . . . . . . 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))))
60	30, 31, 32, 33	ruclem8 14805	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
61	49, 60	syldan 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
62	45, 46, 52, 54, 57, 58, 59, 61	ruclem2 14800	. . . . . . . . 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))))
63	62	simp1d 1066	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
64	30, 31, 32, 33	ruclem7 14804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
65	49, 64	syldan 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
66		1st2nd2 7096	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩)
67	50, 66	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩)
68	67	oveq1d 6564	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
69	65, 68	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
70	69	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘(𝑛 + 1))) = (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
71	63, 70	breqtrrd 4611	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑛)) ≤ (1^st ‘(𝐺‘(𝑛 + 1))))
72	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘𝑀)) ∈ ℝ)
73		peano2nn0 11210	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
74	49, 73	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝑛 + 1) ∈ ℕ₀)
75	47, 74	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
76		xp1st 7089	. . . . . . . . 9 ⊢ ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
77	75, 76	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (1^st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
78		letr 10010	. . . . . . . 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)))))
79	72, 52, 77, 78	syl3anc 1318	. . . . . . 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)))))
80	71, 79	mpan2d 706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑛)) → (1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘(𝑛 + 1)))))
81	69	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) = (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
82	62	simp3d 1068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2^nd ‘(𝐺‘𝑛)))
83	81, 82	eqbrtrd 4605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑛)))
84		xp2nd 7090	. . . . . . . . 9 ⊢ ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
85	75, 84	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
86	41	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (2^nd ‘(𝐺‘𝑀)) ∈ ℝ)
87		letr 10010	. . . . . . . 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))))
88	85, 54, 86, 87	syl3anc 1318	. . . . . . 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))))
89	83, 88	mpand 707	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((2^nd ‘(𝐺‘𝑛)) ≤ (2^nd ‘(𝐺‘𝑀)) → (2^nd ‘(𝐺‘(𝑛 + 1))) ≤ (2^nd ‘(𝐺‘𝑀))))
90	80, 89	anim12d 584	. . . . 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)))))
91	90	expcom 450	. . . 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))))))
92	91	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))))))
93	8, 15, 22, 29, 44, 92	uzind4 11622	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → ((1^st ‘(𝐺‘𝑀)) ≤ (1^st ‘(𝐺‘𝑁)) ∧ (2^nd ‘(𝐺‘𝑁)) ≤ (2^nd ‘(𝐺‘𝑀)))))
94	1, 93	mpcom 37	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 383 = wceq 1475 ∈ wcel 1977 ⦋csb 3499 ∪ cun 3538 ifcif 4036 {csn 4125 ⟨cop 4131 class class class wbr 4583 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 seqcseq 12663

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664

This theorem is referenced by: ruclem10 14807

W3C validator