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Theorem ruclem8 15450

Description: Lemma for ruc 15456. 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 6504	. . . . 5 ⊢ (𝑘 = 0 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘0)))
2		2fveq3 6504	. . . . 5 ⊢ (𝑘 = 0 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘0)))
3	1, 2	breq12d 4942	. . . 4 ⊢ (𝑘 = 0 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0))))
4	3	imbi2d 333	. . 3 ⊢ (𝑘 = 0 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0)))))
5		2fveq3 6504	. . . . 5 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
6		2fveq3 6504	. . . . 5 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
7	5, 6	breq12d 4942	. . . 4 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛))))
8	7	imbi2d 333	. . 3 ⊢ (𝑘 = 𝑛 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))))
9		2fveq3 6504	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
10		2fveq3 6504	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
11	9, 10	breq12d 4942	. . . 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 333	. . 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 6504	. . . . 5 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
14		2fveq3 6504	. . . . 5 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
15	13, 14	breq12d 4942	. . . 4 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁))))
16	15	imbi2d 333	. . 3 ⊢ (𝑘 = 𝑁 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))))
17		0lt1 10963	. . . . 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 15447	. . . . . 6 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
24	23	fveq2d 6503	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = (1^st ‘⟨0, 1⟩))
25		c0ex 10433	. . . . . 6 ⊢ 0 ∈ V
26		1ex 10435	. . . . . 6 ⊢ 1 ∈ V
27	25, 26	op1st 7509	. . . . 5 ⊢ (1^st ‘⟨0, 1⟩) = 0
28	24, 27	syl6eq 2830	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = 0)
29	23	fveq2d 6503	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
30	25, 26	op2nd 7510	. . . . 5 ⊢ (2^nd ‘⟨0, 1⟩) = 1
31	29, 30	syl6eq 2830	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
32	18, 28, 31	3brtr4d 4961	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0)))
33	19	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → 𝐹:ℕ⟶ℝ)
34	20	adantr 473	. . . . . . . . 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 15448	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
36	35	ffvelrnda 6676	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
37	36	adantrr 704	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
38		xp1st 7533	. . . . . . . . . 10 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
39	37, 38	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
40		xp2nd 7534	. . . . . . . . . 10 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
41	37, 40	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
42		nn0p1nn 11748	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
43		ffvelrn 6674	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
44	19, 42, 43	syl2an 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
45	44	adantrr 704	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46		eqid 2778	. . . . . . . . 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 2778	. . . . . . . . 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 760	. . . . . . . . 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 15445	. . . . . . . 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 1123	. . . . . . 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 15449	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
52	51	adantrr 704	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
53		1st2nd2 7540	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐺‘𝑛)), (2^nd ‘(𝐺‘𝑛))⟩)
54	37, 53	syl 17	. . . . . . . . . 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 6991	. . . . . . . . 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 2814	. . . . . . . 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 6503	. . . . . . 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 6503	. . . . . . 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 4961	. . . . . 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 449	. . . . 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 406	. . . 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 29	. . 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 11890	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁))))
64	63	impcom 399	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⦋csb 3786 ∪ cun 3827 ifcif 4350 {csn 4441 ⟨cop 4447 class class class wbr 4929 × cxp 5405 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 1^st c1st 7499 2^nd c2nd 7500 ℝcr 10334 0cc0 10335 1c1 10336 + caddc 10338 < clt 10474 ≤ cle 10475 / cdiv 11098 ℕcn 11439 2c2 11495 ℕ₀cn0 11707 seqcseq 13184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-seq 13185

This theorem is referenced by: ruclem9 15451 ruclem10 15452 ruclem12 15454

W3C validator