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Theorem ruclem8 16328

Description: Lemma for ruc 16334. 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 6884	. . . . 5 ⊢ (𝑘 = 0 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘0)))
2		2fveq3 6884	. . . . 5 ⊢ (𝑘 = 0 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘0)))
3	1, 2	breq12d 5116	. . . 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 6884	. . . . 5 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
6		2fveq3 6884	. . . . 5 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
7	5, 6	breq12d 5116	. . . 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 6884	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
10		2fveq3 6884	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
11	9, 10	breq12d 5116	. . . 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 6884	. . . . 5 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
14		2fveq3 6884	. . . . 5 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
15	13, 14	breq12d 5116	. . . 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 11763	. . . . 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 16325	. . . . . 6 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
24	23	fveq2d 6883	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = (1^st ‘⟨0, 1⟩))
25		c0ex 11227	. . . . . 6 ⊢ 0 ∈ V
26		1ex 11230	. . . . . 6 ⊢ 1 ∈ V
27	25, 26	op1st 7995	. . . . 5 ⊢ (1^st ‘⟨0, 1⟩) = 0
28	24, 27	eqtrdi 2811	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = 0)
29	23	fveq2d 6883	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
30	25, 26	op2nd 7996	. . . . 5 ⊢ (2^nd ‘⟨0, 1⟩) = 1
31	29, 30	eqtrdi 2811	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
32	18, 28, 31	3brtr4d 5137	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0)))
33	19	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → 𝐹:ℕ⟶ℝ)
34	20	adantr 486	. . . . . . . . 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 16326	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
36	35	ffvelcdmda 7078	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
37	36	adantrr 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
38		xp1st 8019	. . . . . . . . . 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 8020	. . . . . . . . . 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 12570	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
43		ffvelcdm 7075	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
44	19, 42, 43	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
45	44	adantrr 730	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46		eqid 2760	. . . . . . . . 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 2760	. . . . . . . . 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 785	. . . . . . . . 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 16323	. . . . . . . 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 1161	. . . . . . 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 16327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
52	51	adantrr 730	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
53		1st2nd2 8026	. . . . . . . . . . 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 7429	. . . . . . . . 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 2795	. . . . . . . 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 6883	. . . . . . 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 6883	. . . . . . 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 5137	. . . . . 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 462	. . . . 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 419	. . . 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 12719	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁))))
64	63	impcom 413	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⦋csb 3847 ∪ cun 3897 ifcif 4482 {csn 4584 ⟨cop 4590 class class class wbr 5103 × cxp 5653 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ∈ cmpo 7416 1^st c1st 7985 2^nd c2nd 7986 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 < clt 11270 ≤ cle 11271 / cdiv 11898 ℕcn 12260 2c2 12322 ℕ₀cn0 12531 seqcseq 14068

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13565 df-seq 14069

This theorem is used by: ruclem9 16329 ruclem10 16330 ruclem12 16332

W3C validator