Theorem ruclem29 4913

Description: Lemma for ruc 4924. At any index A, the interval between our constructed functions G and H does not include the corresponding value of input function F. In other words, our constructed functions define, by ruclem26 4910 and ruclem27 4911, an ever-shrinking interval that eventually squeezes out all values of F.

Hypotheses

Ref	Expression
ruclem.0	⊢ F:ℕ–→ℝ
ruclem.1	⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
ruclem.2	⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}
ruclem.3	⊢ G = (1st ∘ (DseqC))
ruclem.4	⊢ H = (2nd ∘ (DseqC))
ruclem28.a	⊢ A ∈ ℕ

Assertion

Ref	Expression
ruclem29	⊢ ¬ ((G ‘A) < (F ‘A) ∧ (F ‘A) < (H ‘A))

Distinct variable group(s):   x,y,z   z,F

Proof of Theorem ruclem29

Step	Hyp	Ref	Expression
1		ruclem28.a	. 2 ⊢ A ∈ ℕ
2		fveq2 2832	. . . . . 6 ⊢ (w = 1 → (G ‘w) = (G ‘1))
3		fveq2 2832	. . . . . 6 ⊢ (w = 1 → (F ‘w) = (F ‘1))
4	2, 3	breq12d 2073	. . . . 5 ⊢ (w = 1 → ((G ‘w) < (F ‘w) ↔ (G ‘1) < (F ‘1)))
5		fveq2 2832	. . . . . 6 ⊢ (w = 1 → (H ‘w) = (H ‘1))
6	3, 5	breq12d 2073	. . . . 5 ⊢ (w = 1 → ((F ‘w) < (H ‘w) ↔ (F ‘1) < (H ‘1)))
7	4, 6	anbi12d 476	. . . 4 ⊢ (w = 1 → (((G ‘w) < (F ‘w) ∧ (F ‘w) < (H ‘w)) ↔ ((G ‘1) < (F ‘1) ∧ (F ‘1) < (H ‘1))))
8	7	negbid 463	. . 3 ⊢ (w = 1 → (¬ ((G ‘w) < (F ‘w) ∧ (F ‘w) < (H ‘w)) ↔ ¬ ((G ‘1) < (F ‘1) ∧ (F ‘1) < (H ‘1))))
9		fveq2 2832	. . . . . 6 ⊢ (w = (v + 1) → (G ‘w) = (G ‘(v + 1)))
10		fveq2 2832	. . . . . 6 ⊢ (w = (v + 1) → (F ‘w) = (F ‘(v + 1)))
11	9, 10	breq12d 2073	. . . . 5 ⊢ (w = (v + 1) → ((G ‘w) < (F ‘w) ↔ (G ‘(v + 1)) < (F ‘(v + 1))))
12		fveq2 2832	. . . . . 6 ⊢ (w = (v + 1) → (H ‘w) = (H ‘(v + 1)))
13	10, 12	breq12d 2073	. . . . 5 ⊢ (w = (v + 1) → ((F ‘w) < (H ‘w) ↔ (F ‘(v + 1)) < (H ‘(v + 1))))
14	11, 13	anbi12d 476	. . . 4 ⊢ (w = (v + 1) → (((G ‘w) < (F ‘w) ∧ (F ‘w) < (H ‘w)) ↔ ((G ‘(v + 1)) < (F ‘(v + 1)) ∧ (F ‘(v + 1)) < (H ‘(v + 1)))))
15	14	negbid 463	. . 3 ⊢ (w = (v + 1) → (¬ ((G ‘w) < (F ‘w) ∧ (F ‘w) < (H ‘w)) ↔ ¬ ((G ‘(v + 1)) < (F ‘(v + 1)) ∧ (F ‘(v + 1)) < (H ‘(v + 1)))))
16		fveq2 2832	. . . . . 6 ⊢ (w = A → (G ‘w) = (G ‘A))
17		fveq2 2832	. . . . . 6 ⊢ (w = A → (F ‘w) = (F ‘A))
18	16, 17	breq12d 2073	. . . . 5 ⊢ (w = A → ((G ‘w) < (F ‘w) ↔ (G ‘A) < (F ‘A)))
19		fveq2 2832	. . . . . 6 ⊢ (w = A → (H ‘w) = (H ‘A))
20	17, 19	breq12d 2073	. . . . 5 ⊢ (w = A → ((F ‘w) < (H ‘w) ↔ (F ‘A) < (H ‘A)))
21	18, 20	anbi12d 476	. . . 4 ⊢ (w = A → (((G ‘w) < (F ‘w) ∧ (F ‘w) < (H ‘w)) ↔ ((G ‘A) < (F ‘A) ∧ (F ‘A) < (H ‘A))))
22	21	negbid 463	. . 3 ⊢ (w = A → (¬ ((G ‘w) < (F ‘w) ∧ (F ‘w) < (H ‘w)) ↔ ¬ ((G ‘A) < (F ‘A) ∧ (F ‘A) < (H ‘A))))
23		ruclem.0	. . . . . . . 8 ⊢ F:ℕ–→ℝ
24		1nn 4432	. . . . . . . 8 ⊢ 1 ∈ ℕ
25		ffvrn 2890	. . . . . . . 8 ⊢ ((F:ℕ–→ℝ ∧ 1 ∈ ℕ) → (F ‘1) ∈ ℝ)
26	23, 24, 25	mp2an 520	. . . . . . 7 ⊢ (F ‘1) ∈ ℝ
27	26	ltplus1 4384	. . . . . 6 ⊢ (F ‘1) < ((F ‘1) + 1)
28		ruclem.1	. . . . . . 7 ⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
29		ruclem.2	. . . . . . 7 ⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}
30		ruclem.3	. . . . . . 7 ⊢ G = (1st ∘ (DseqC))
31		ruclem.4	. . . . . . 7 ⊢ H = (2nd ∘ (DseqC))
32	23, 28, 29, 30, 31	ruclem16 4900	. . . . . 6 ⊢ (G ‘1) = ((F ‘1) + 1)
33	27, 32	breqtrr 2082	. . . . 5 ⊢ (F ‘1) < (G ‘1)
34	23, 28, 29, 30, 31, 24	ruclem22 4906	. . . . . 6 ⊢ (G ‘1) ∈ ℝ
35	26, 34	ltnsym 4300	. . . . 5 ⊢ ((F ‘1) < (G ‘1) → ¬ (G ‘1) < (F ‘1))
36	33, 35	ax-mp 6	. . . 4 ⊢ ¬ (G ‘1) < (F ‘1)
37	36	intnanr 517	. . 3 ⊢ ¬ ((G ‘1) < (F ‘1) ∧ (F ‘1) < (H ‘1))
38		opreq1 3006	. . . . . . . 8 ⊢ (v = if(v ∈ ℕ, v, 1) → (v + 1) = (if(v ∈ ℕ, v, 1) + 1))
39	38	fveq2d 2836	. . . . . . 7 ⊢ (v = if(v ∈ ℕ, v, 1) → (G ‘(v + 1)) = (G ‘(if(v ∈ ℕ, v, 1) + 1)))
40	38	fveq2d 2836	. . . . . . 7 ⊢ (v = if(v ∈ ℕ, v, 1) → (F ‘(v + 1)) = (F ‘(if(v ∈ ℕ, v, 1) + 1)))
41	39, 40	breq12d 2073	. . . . . 6 ⊢ (v = if(v ∈ ℕ, v, 1) → ((G ‘(v + 1)) < (F ‘(v + 1)) ↔ (G ‘(if(v ∈ ℕ, v, 1) + 1)) < (F ‘(if(v ∈ ℕ, v, 1) + 1))))
42	38	fveq2d 2836	. . . . . . 7 ⊢ (v = if(v ∈ ℕ, v, 1) → (H ‘(v + 1)) = (H ‘(if(v ∈ ℕ, v, 1) + 1)))
43	40, 42	breq12d 2073	. . . . . 6 ⊢ (v = if(v ∈ ℕ, v, 1) → ((F ‘(v + 1)) < (H ‘(v + 1)) ↔ (F ‘(if(v ∈ ℕ, v, 1) + 1)) < (H ‘(if(v ∈ ℕ, v, 1) + 1))))
44	41, 43	anbi12d 476	. . . . 5 ⊢ (v = if(v ∈ ℕ, v, 1) → (((G ‘(v + 1)) < (F ‘(v + 1)) ∧ (F ‘(v + 1)) < (H ‘(v + 1))) ↔ ((G ‘(if(v ∈ ℕ, v, 1) + 1)) < (F ‘(if(v ∈ ℕ, v, 1) + 1)) ∧ (F ‘(if(v ∈ ℕ, v, 1) + 1)) < (H ‘(if(v ∈ ℕ, v, 1) + 1)))))
45	44	negbid 463	. . . 4 ⊢ (v = if(v ∈ ℕ, v, 1) → (¬ ((G ‘(v + 1)) < (F ‘(v + 1)) ∧ (F ‘(v + 1)) < (H ‘(v + 1))) ↔ ¬ ((G ‘(if(v ∈ ℕ, v, 1) + 1)) < (F ‘(if(v ∈ ℕ, v, 1) + 1)) ∧ (F ‘(if(v ∈ ℕ, v, 1) + 1)) < (H ‘(if(v ∈ ℕ, v, 1) + 1)))))
46	24	elimel 1793	. . . . 5 ⊢ if(v ∈ ℕ, v, 1) ∈ ℕ
47	23, 28, 29, 30, 31, 46	ruclem28 4912	. . . 4 ⊢ ¬ ((G ‘(if(v ∈ ℕ, v, 1) + 1)) < (F ‘(if(v ∈ ℕ, v, 1) + 1)) ∧ (F ‘(if(v ∈ ℕ, v, 1) + 1)) < (H ‘(if(v ∈ ℕ, v, 1) + 1)))
48	45, 47	dedth 1784	. . 3 ⊢ (v ∈ ℕ → ¬ ((G ‘(v + 1)) < (F ‘(v + 1)) ∧ (F ‘(v + 1)) < (H ‘(v + 1))))
49	8, 15, 22, 37, 48	nn1suc 4435	. 2 ⊢ (A ∈ ℕ → ¬ ((G ‘A) < (F ‘A) ∧ (F ‘A) < (H ‘A)))
50	1, 49	ax-mp 6	1 ⊢ ¬ ((G ‘A) < (F ‘A) ∧ (F ‘A) < (H ‘A))

Syntax hints:  ¬ wn 1   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∪ cun 1485  ifcif 1776  {csn 1808  ⟨cop 1810   class class class wbr 2054   × cxp 2408   ↾ cres 2412   ∘ ccom 2414  –→wf 2418   ‘cfv 2422  (class class class)co 3001  {copab2 3002  1^st c1st 3085  2^nd c2nd 3086  ℝcr 4027  1c1 4029   + caddc 4031   · cmulc 4032   < clt 4033   / cdiv 4091  ℕcn 4093  2c2 4454  3c3 4455  seqcseq 4660

This theorem is referenced by:  ruclem36 4920

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-n0 4535  df-z 4564  df-seq 4661

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