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Theorem ruclem8 16146

Description: Lemma for ruc 16152. 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 6827	. . . . 5 ⊢ (𝑘 = 0 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘0)))
2		2fveq3 6827	. . . . 5 ⊢ (𝑘 = 0 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘0)))
3	1, 2	breq12d 5102	. . . 4 ⊢ (𝑘 = 0 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0))))
4	3	imbi2d 340	. . 3 ⊢ (𝑘 = 0 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0)))))
5		2fveq3 6827	. . . . 5 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑛)))
6		2fveq3 6827	. . . . 5 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑛)))
7	5, 6	breq12d 5102	. . . 4 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛))))
8	7	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑛 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))))
9		2fveq3 6827	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘(𝑛 + 1))))
10		2fveq3 6827	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘(𝑛 + 1))))
11	9, 10	breq12d 5102	. . . 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 340	. . 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 6827	. . . . 5 ⊢ (𝑘 = 𝑁 → (1^st ‘(𝐺‘𝑘)) = (1^st ‘(𝐺‘𝑁)))
14		2fveq3 6827	. . . . 5 ⊢ (𝑘 = 𝑁 → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘(𝐺‘𝑁)))
15	13, 14	breq12d 5102	. . . 4 ⊢ (𝑘 = 𝑁 → ((1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘)) ↔ (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁))))
16	15	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑁 → ((𝜑 → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑘))) ↔ (𝜑 → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))))
17		0lt1 11639	. . . . 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 16143	. . . . . 6 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
24	23	fveq2d 6826	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = (1^st ‘⟨0, 1⟩))
25		c0ex 11106	. . . . . 6 ⊢ 0 ∈ V
26		1ex 11108	. . . . . 6 ⊢ 1 ∈ V
27	25, 26	op1st 7929	. . . . 5 ⊢ (1^st ‘⟨0, 1⟩) = 0
28	24, 27	eqtrdi 2782	. . . 4 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) = 0)
29	23	fveq2d 6826	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
30	25, 26	op2nd 7930	. . . . 5 ⊢ (2^nd ‘⟨0, 1⟩) = 1
31	29, 30	eqtrdi 2782	. . . 4 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
32	18, 28, 31	3brtr4d 5121	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺‘0)) < (2^nd ‘(𝐺‘0)))
33	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → 𝐹:ℕ⟶ℝ)
34	20	adantr 480	. . . . . . . . 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 16144	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
36	35	ffvelcdmda 7017	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
37	36	adantrr 717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
38		xp1st 7953	. . . . . . . . . 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 7954	. . . . . . . . . 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 12420	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
43		ffvelcdm 7014	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
44	19, 42, 43	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
45	44	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46		eqid 2731	. . . . . . . . 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 2731	. . . . . . . . 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 772	. . . . . . . . 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 16141	. . . . . . . 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 1143	. . . . . . 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 16145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
52	51	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
53		1st2nd2 7960	. . . . . . . . . . 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 7361	. . . . . . . . 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 2766	. . . . . . . 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 6826	. . . . . . 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 6826	. . . . . . 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 5121	. . . . . 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 456	. . . . 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 413	. . . 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 12568	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁))))
64	63	impcom 407	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑁)) < (2^nd ‘(𝐺‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⦋csb 3845 ∪ cun 3895 ifcif 4472 {csn 4573 ⟨cop 4579 class class class wbr 5089 × cxp 5612 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 < clt 11146 ≤ cle 11147 / cdiv 11774 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 seqcseq 13908

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-seq 13909

This theorem is referenced by: ruclem9 16147 ruclem10 16148 ruclem12 16150

W3C validator