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Theorem ruclem8 16293

Description: Lemma for ruc 16299. The intervals of the 𝐺 sequence are all nonempty. (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)

**Assertion**
Ref	Expression
ruclem8	⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))

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

Proof of Theorem ruclem8 Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6887	. . . . 5 ⊢ (𝑘 = 0 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘0)))
2		2fveq3 6887	. . . . 5 ⊢ (𝑘 = 0 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘0)))
3	1, 2	breq12d 5126	. . . 4 ⊢ (𝑘 = 0 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0))))
4	3	imbi2d 343	. . 3 ⊢ (𝑘 = 0 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0)))))
5		2fveq3 6887	. . . . 5 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
6		2fveq3 6887	. . . . 5 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
7	5, 6	breq12d 5126	. . . 4 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛))))
8	7	imbi2d 343	. . 3 ⊢ (𝑘 = 𝑛 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))))
9		2fveq3 6887	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
10		2fveq3 6887	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
11	9, 10	breq12d 5126	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘(𝑛 + 1))) < (2^nd ‘(𝐺‘(𝑛 + 1)))))
12	11	imbi2d 343	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘(𝑛 + 1))) < (2^nd ‘(𝐺‘(𝑛 + 1))))))
13		2fveq3 6887	. . . . 5 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
14		2fveq3 6887	. . . . 5 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
15	13, 14	breq12d 5126	. . . 4 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁))))
16	15	imbi2d 343	. . 3 ⊢ (𝑘 = 𝑁 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))))
17		0lt1 11736	. . . . 5 ⊢ 0 < 1
18	17	a1i 11	. . . 4 ⊢ (𝜑 → 0 < 1)
19		ruc.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
20		ruc.2	. . . . . . 7 ⊢ (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)⟩)))
21		ruc.4	. . . . . . 7 ⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
22		ruc.5	. . . . . . 7 ⊢ 𝐺 = seq0(𝐷, 𝐶)
23	19, 20, 21, 22	ruclem4 16290	. . . . . 6 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
24	23	fveq2d 6886	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = (1^st ‘⟨0, 1⟩))
25		c0ex 11200	. . . . . 6 ⊢ 0 ∈ V
26		1ex 11203	. . . . . 6 ⊢ 1 ∈ V
27	25, 26	op1st 7994	. . . . 5 ⊢ (1^st ‘⟨0, 1⟩) = 0
28	24, 27	eqtrdi 2820	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = 0)
29	23	fveq2d 6886	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
30	25, 26	op2nd 7995	. . . . 5 ⊢ (2^nd ‘⟨0, 1⟩) = 1
31	29, 30	eqtrdi 2820	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
32	18, 28, 31	3brtr4d 5147	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0)))
33	19	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → 𝐹:ℕ⟶ℝ)
34	20	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → 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)⟩)))
35	19, 20, 21, 22	ruclem6 16291	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
36	35	ffvelcdmda 7080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
37	36	adantrr 729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
38		xp1st 8018	. . . . . . . . . 10 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
39	37, 38	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
40		xp2nd 8019	. . . . . . . . . 10 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
41	37, 40	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
42		nn0p1nn 12543	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
43		ffvelcdm 7077	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
44	19, 42, 43	syl2an 607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
45	44	adantrr 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46		eqid 2769	. . . . . . . . 9 ⊢ (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))))
47		eqid 2769	. . . . . . . . 9 ⊢ (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))))
48		simprr 784	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))
49	33, 34, 39, 41, 45, 46, 47, 48	ruclem2 16288	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → ((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))))
50	49	simp2d 1159	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (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)))))
51	19, 20, 21, 22	ruclem7 16292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
52	51	adantrr 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
53		1st2nd2 8025	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩)
54	37, 53	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩)
55	54	oveq1d 7426	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
56	52, 55	eqtrd 2804	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘(𝑛 + 1)) = (⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
57	56	fveq2d 6886	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (1^st ‘(𝐺‘(𝑛 + 1))) = (1^st ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
58	56	fveq2d 6886	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (2^nd ‘(𝐺‘(𝑛 + 1))) = (2^nd ‘(⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
59	50, 57, 58	3brtr4d 5147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (1^st ‘(𝐺‘(𝑛 + 1))) < (2^nd ‘(𝐺‘(𝑛 + 1))))
60	59	expr 461	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)) → (1^st ‘(𝐺‘(𝑛 + 1))) < (2^nd ‘(𝐺‘(𝑛 + 1)))))
61	60	expcom 418	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (𝜑 → ((1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)) → (1^st ‘(𝐺‘(𝑛 + 1))) < (2^nd ‘(𝐺‘(𝑛 + 1))))))
62	61	a2d 30	. . 3 ⊢ (𝑛 ∈ ℕ₀ → ((𝜑 → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛))) → (𝜑 → (1^st ‘(𝐺‘(𝑛 + 1))) < (2^nd ‘(𝐺‘(𝑛 + 1))))))
63	4, 8, 12, 16, 32, 62	nn0ind 12691	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁))))
64	63	impcom 412	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ⦋csb 3861 ∪ cun 3911 ifcif 4492 {csn 4594 ⟨cop 4600 class class class wbr 5113 × cxp 5660 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7984 2^nd c2nd 7985 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 < clt 11243 ≤ cle 11244 / cdiv 11871 ℕcn 12233 2c2 12295 ℕ₀cn0 12504 seqcseq 14037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-n0 12505 df-z 12592 df-uz 12863 df-fz 13536 df-seq 14038

This theorem is referenced by: ruclem9 16294 ruclem10 16295 ruclem12 16297

W3C validator