Theorem ruclem35 4919

Description: Lemma for ruc 4924. The supremum we have constructed lies between all values of the G and H functions. Compare ruclem29 4913, which states the opposite for the input function 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))
ruclem.5	⊢ S = sup(ran G, ℝ, < )
ruclem.a	⊢ A ∈ ℕ

Assertion

Ref	Expression
ruclem35	⊢ ((G ‘A) < S ∧ S < (H ‘A))

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

Proof of Theorem ruclem35

Step	Hyp	Ref	Expression
1		ruclem.0	. . . 4 ⊢ F:ℕ–→ℝ
2		ruclem.1	. . . 4 ⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
3		ruclem.2	. . . 4 ⊢ 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)⟩))}
4		ruclem.3	. . . 4 ⊢ G = (1st ∘ (DseqC))
5		ruclem.4	. . . 4 ⊢ H = (2nd ∘ (DseqC))
6		ruclem.a	. . . 4 ⊢ A ∈ ℕ
7	1, 2, 3, 4, 5, 6	ruclem26 4910	. . 3 ⊢ (G ‘A) < (G ‘(A + 1))
8	1, 2, 3, 4, 5	ruclem17 4901	. . . . . . 7 ⊢ G:ℕ–→ℝ
9		ffn 2752	. . . . . . 7 ⊢ (G:ℕ–→ℝ → G Fn ℕ)
10	8, 9	ax-mp 6	. . . . . 6 ⊢ G Fn ℕ
11		peano2nn 4433	. . . . . . 7 ⊢ (A ∈ ℕ → (A + 1) ∈ ℕ)
12	6, 11	ax-mp 6	. . . . . 6 ⊢ (A + 1) ∈ ℕ
13		fnfvrn 2889	. . . . . 6 ⊢ ((G Fn ℕ ∧ (A + 1) ∈ ℕ) → (G ‘(A + 1)) ∈ ran G)
14	10, 12, 13	mp2an 520	. . . . 5 ⊢ (G ‘(A + 1)) ∈ ran G
15	1, 2, 3, 4, 5	ruclem33 4917	. . . . . 6 ⊢ (ran G ⊆ ℝ ∧ ¬ ran G = ∅ ∧ ∃w ∈ ℝ ∀v ∈ ran Gv ≤ w)
16	15	suprubi 4517	. . . . 5 ⊢ ((G ‘(A + 1)) ∈ ran G → (G ‘(A + 1)) ≤ sup(ran G, ℝ, < ))
17	14, 16	ax-mp 6	. . . 4 ⊢ (G ‘(A + 1)) ≤ sup(ran G, ℝ, < )
18		ruclem.5	. . . 4 ⊢ S = sup(ran G, ℝ, < )
19	17, 18	breqtrr 2082	. . 3 ⊢ (G ‘(A + 1)) ≤ S
20	1, 2, 3, 4, 5, 6	ruclem22 4906	. . . 4 ⊢ (G ‘A) ∈ ℝ
21	1, 2, 3, 4, 5, 12	ruclem22 4906	. . . 4 ⊢ (G ‘(A + 1)) ∈ ℝ
22	1, 2, 3, 4, 5, 18	ruclem34 4918	. . . 4 ⊢ S ∈ ℝ
23	20, 21, 22	ltletr 4309	. . 3 ⊢ (((G ‘A) < (G ‘(A + 1)) ∧ (G ‘(A + 1)) ≤ S) → (G ‘A) < S)
24	7, 19, 23	mp2an 520	. 2 ⊢ (G ‘A) < S
25	1, 2, 3, 4, 5, 12	ruclem23 4907	. . . . . 6 ⊢ (H ‘(A + 1)) ∈ ℝ
26		fvelrn 2883	. . . . . . . . 9 ⊢ (G Fn ℕ → (u ∈ ran G ↔ ∃w ∈ ℕ (G ‘w) = u))
27	10, 26	ax-mp 6	. . . . . . . 8 ⊢ (u ∈ ran G ↔ ∃w ∈ ℕ (G ‘w) = u)
28		breq2 2066	. . . . . . . . . . . 12 ⊢ ((G ‘w) = u → ((H ‘(A + 1)) < (G ‘w) ↔ (H ‘(A + 1)) < u))
29	28	negbid 463	. . . . . . . . . . 11 ⊢ ((G ‘w) = u → (¬ (H ‘(A + 1)) < (G ‘w) ↔ ¬ (H ‘(A + 1)) < u))
30		ltnsymt 4294	. . . . . . . . . . . 12 ⊢ (((G ‘w) ∈ ℝ ∧ (H ‘(A + 1)) ∈ ℝ) → ((G ‘w) < (H ‘(A + 1)) → ¬ (H ‘(A + 1)) < (G ‘w)))
31		fveq2 2832	. . . . . . . . . . . . . . 15 ⊢ (w = if(w ∈ ℕ, w, 1) → (G ‘w) = (G ‘if(w ∈ ℕ, w, 1)))
32	31	eleq1d 1155	. . . . . . . . . . . . . 14 ⊢ (w = if(w ∈ ℕ, w, 1) → ((G ‘w) ∈ ℝ ↔ (G ‘if(w ∈ ℕ, w, 1)) ∈ ℝ))
33		1nn 4432	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
34	33	elimel 1793	. . . . . . . . . . . . . . 15 ⊢ if(w ∈ ℕ, w, 1) ∈ ℕ
35	1, 2, 3, 4, 5, 34	ruclem22 4906	. . . . . . . . . . . . . 14 ⊢ (G ‘if(w ∈ ℕ, w, 1)) ∈ ℝ
36	32, 35	dedth 1784	. . . . . . . . . . . . 13 ⊢ (w ∈ ℕ → (G ‘w) ∈ ℝ)
37	36, 25	jctir 241	. . . . . . . . . . . 12 ⊢ (w ∈ ℕ → ((G ‘w) ∈ ℝ ∧ (H ‘(A + 1)) ∈ ℝ))
38	31	breq1d 2071	. . . . . . . . . . . . 13 ⊢ (w = if(w ∈ ℕ, w, 1) → ((G ‘w) < (H ‘(A + 1)) ↔ (G ‘if(w ∈ ℕ, w, 1)) < (H ‘(A + 1))))
39	1, 2, 3, 4, 5, 34, 12	ruclem32 4916	. . . . . . . . . . . . 13 ⊢ (G ‘if(w ∈ ℕ, w, 1)) < (H ‘(A + 1))
40	38, 39	dedth 1784	. . . . . . . . . . . 12 ⊢ (w ∈ ℕ → (G ‘w) < (H ‘(A + 1)))
41	30, 37, 40	sylc 62	. . . . . . . . . . 11 ⊢ (w ∈ ℕ → ¬ (H ‘(A + 1)) < (G ‘w))
42	29, 41	syl5bi 183	. . . . . . . . . 10 ⊢ ((G ‘w) = u → (w ∈ ℕ → ¬ (H ‘(A + 1)) < u))
43	42	com12 13	. . . . . . . . 9 ⊢ (w ∈ ℕ → ((G ‘w) = u → ¬ (H ‘(A + 1)) < u))
44	43	r19.23aiv 1284	. . . . . . . 8 ⊢ (∃w ∈ ℕ (G ‘w) = u → ¬ (H ‘(A + 1)) < u)
45	27, 44	sylbi 174	. . . . . . 7 ⊢ (u ∈ ran G → ¬ (H ‘(A + 1)) < u)
46	45	rgen 1247	. . . . . 6 ⊢ ∀u ∈ ran G ¬ (H ‘(A + 1)) < u
47	15	suprnubi 4519	. . . . . 6 ⊢ (((H ‘(A + 1)) ∈ ℝ ∧ ∀u ∈ ran G ¬ (H ‘(A + 1)) < u) → ¬ (H ‘(A + 1)) < sup(ran G, ℝ, < ))
48	25, 46, 47	mp2an 520	. . . . 5 ⊢ ¬ (H ‘(A + 1)) < sup(ran G, ℝ, < )
49	18	breq2i 2069	. . . . . 6 ⊢ ((H ‘(A + 1)) < S ↔ (H ‘(A + 1)) < sup(ran G, ℝ, < ))
50	49	negbii 162	. . . . 5 ⊢ (¬ (H ‘(A + 1)) < S ↔ ¬ (H ‘(A + 1)) < sup(ran G, ℝ, < ))
51	48, 50	mpbir 165	. . . 4 ⊢ ¬ (H ‘(A + 1)) < S
52	22, 25	lelt 4301	. . . 4 ⊢ (S ≤ (H ‘(A + 1)) ↔ ¬ (H ‘(A + 1)) < S)
53	51, 52	mpbir 165	. . 3 ⊢ S ≤ (H ‘(A + 1))
54	1, 2, 3, 4, 5, 6	ruclem27 4911	. . 3 ⊢ (H ‘(A + 1)) < (H ‘A)
55	1, 2, 3, 4, 5, 6	ruclem23 4907	. . . 4 ⊢ (H ‘A) ∈ ℝ
56	22, 25, 55	lelttr 4308	. . 3 ⊢ ((S ≤ (H ‘(A + 1)) ∧ (H ‘(A + 1)) < (H ‘A)) → S < (H ‘A))
57	53, 54, 56	mp2an 520	. 2 ⊢ S < (H ‘A)
58	24, 57	pm3.2i 234	1 ⊢ ((G ‘A) < S ∧ S < (H ‘A))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∖ cdif 1484   ∪ cun 1485  ifcif 1776  {csn 1808  ⟨cop 1810   class class class wbr 2054  supcsup 2060   × cxp 2408  ran crn 2411   ↾ cres 2412   ∘ ccom 2414   Fn wfn 2417  –→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   ≤ cle 4092  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-sup 2154  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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