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Theorem ruclem8 16317

Description: Lemma for ruc 16323. 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 6890	. . . . 5 ⊢ (𝑘 = 0 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘0)))
2		2fveq3 6890	. . . . 5 ⊢ (𝑘 = 0 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘0)))
3	1, 2	breq12d 5124	. . . 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 6890	. . . . 5 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
6		2fveq3 6890	. . . . 5 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
7	5, 6	breq12d 5124	. . . 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 6890	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
10		2fveq3 6890	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
11	9, 10	breq12d 5124	. . . 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 6890	. . . . 5 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
14		2fveq3 6890	. . . . 5 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
15	13, 14	breq12d 5124	. . . 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 11753	. . . . 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 16314	. . . . . 6 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
24	23	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = (1^st ‘⟨0, 1⟩))
25		c0ex 11217	. . . . . 6 ⊢ 0 ∈ V
26		1ex 11220	. . . . . 6 ⊢ 1 ∈ V
27	25, 26	op1st 8000	. . . . 5 ⊢ (1^st ‘⟨0, 1⟩) = 0
28	24, 27	eqtrdi 2816	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = 0)
29	23	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
30	25, 26	op2nd 8001	. . . . 5 ⊢ (2^nd ‘⟨0, 1⟩) = 1
31	29, 30	eqtrdi 2816	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
32	18, 28, 31	3brtr4d 5145	. . 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 16315	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
36	35	ffvelcdmda 7083	. . . . . . . . . . 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 8024	. . . . . . . . . 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 8025	. . . . . . . . . 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 12560	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
43		ffvelcdm 7080	. . . . . . . . . . 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 2765	. . . . . . . . 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 2765	. . . . . . . . 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 16312	. . . . . . . 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 16316	. . . . . . . . . 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 8031	. . . . . . . . . . 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 7434	. . . . . . . . 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 2800	. . . . . . . 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 6889	. . . . . . 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 6889	. . . . . . 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 5145	. . . . . 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 12709	. 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 2146 ⦋csb 3854 ∪ cun 3904 ifcif 4489 {csn 4591 ⟨cop 4597 class class class wbr 5111 × cxp 5661 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 1^st c1st 7990 2^nd c2nd 7991 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 < clt 11260 ≤ cle 11261 / cdiv 11888 ℕcn 12250 2c2 12312 ℕ₀cn0 12521 seqcseq 14057

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-n0 12522 df-z 12609 df-uz 12881 df-fz 13554 df-seq 14058

This theorem is used by: ruclem9 16318 ruclem10 16319 ruclem12 16321

W3C validator