Theorem ruclem33 4917

Description: Lemma for ruc 4924. The set of values of our constructed function G is a non-empty subset of ℝ. This is a helper lemma for theorems involving supremum.

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))

Assertion

Ref	Expression
ruclem33	⊢ (ran G ⊆ ℝ ∧ ¬ ran G = ∅ ∧ ∃w ∈ ℝ ∀v ∈ ran Gv ≤ w)

Distinct variable group(s):   x,y,z,w,v   w,C,v   w,D,v   w,G,v   w,H,v   z,F,w,v

Proof of Theorem ruclem33

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	1, 2, 3, 4, 5	ruclem17 4901	. . 3 ⊢ G:ℕ–→ℝ
7		frn 2757	. . 3 ⊢ (G:ℕ–→ℝ → ran G ⊆ ℝ)
8	6, 7	ax-mp 6	. 2 ⊢ ran G ⊆ ℝ
9		ffn 2752	. . . . 5 ⊢ (G:ℕ–→ℝ → G Fn ℕ)
10	6, 9	ax-mp 6	. . . 4 ⊢ G Fn ℕ
11		1nn 4432	. . . 4 ⊢ 1 ∈ ℕ
12		fnfvrn 2889	. . . 4 ⊢ ((G Fn ℕ ∧ 1 ∈ ℕ) → (G ‘1) ∈ ran G)
13	10, 11, 12	mp2an 520	. . 3 ⊢ (G ‘1) ∈ ran G
14		n0i 1712	. . 3 ⊢ ((G ‘1) ∈ ran G → ¬ ran G = ∅)
15	13, 14	ax-mp 6	. 2 ⊢ ¬ ran G = ∅
16	1, 2, 3, 4, 5, 11	ruclem23 4907	. . 3 ⊢ (H ‘1) ∈ ℝ
17		fvelrn 2883	. . . . . 6 ⊢ (G Fn ℕ → (v ∈ ran G ↔ ∃w ∈ ℕ (G ‘w) = v))
18	10, 17	ax-mp 6	. . . . 5 ⊢ (v ∈ ran G ↔ ∃w ∈ ℕ (G ‘w) = v)
19		breq1 2065	. . . . . . . 8 ⊢ ((G ‘w) = v → ((G ‘w) ≤ (H ‘1) ↔ v ≤ (H ‘1)))
20		ltlet 4286	. . . . . . . . 9 ⊢ (((G ‘w) ∈ ℝ ∧ (H ‘1) ∈ ℝ) → ((G ‘w) < (H ‘1) → (G ‘w) ≤ (H ‘1)))
21		fveq2 2832	. . . . . . . . . . . 12 ⊢ (w = if(w ∈ ℕ, w, 1) → (G ‘w) = (G ‘if(w ∈ ℕ, w, 1)))
22	21	eleq1d 1155	. . . . . . . . . . 11 ⊢ (w = if(w ∈ ℕ, w, 1) → ((G ‘w) ∈ ℝ ↔ (G ‘if(w ∈ ℕ, w, 1)) ∈ ℝ))
23	11	elimel 1793	. . . . . . . . . . . 12 ⊢ if(w ∈ ℕ, w, 1) ∈ ℕ
24	1, 2, 3, 4, 5, 23	ruclem22 4906	. . . . . . . . . . 11 ⊢ (G ‘if(w ∈ ℕ, w, 1)) ∈ ℝ
25	22, 24	dedth 1784	. . . . . . . . . 10 ⊢ (w ∈ ℕ → (G ‘w) ∈ ℝ)
26	25, 16	jctir 241	. . . . . . . . 9 ⊢ (w ∈ ℕ → ((G ‘w) ∈ ℝ ∧ (H ‘1) ∈ ℝ))
27	21	breq1d 2071	. . . . . . . . . 10 ⊢ (w = if(w ∈ ℕ, w, 1) → ((G ‘w) < (H ‘1) ↔ (G ‘if(w ∈ ℕ, w, 1)) < (H ‘1)))
28	1, 2, 3, 4, 5, 23, 11	ruclem32 4916	. . . . . . . . . 10 ⊢ (G ‘if(w ∈ ℕ, w, 1)) < (H ‘1)
29	27, 28	dedth 1784	. . . . . . . . 9 ⊢ (w ∈ ℕ → (G ‘w) < (H ‘1))
30	20, 26, 29	sylc 62	. . . . . . . 8 ⊢ (w ∈ ℕ → (G ‘w) ≤ (H ‘1))
31	19, 30	syl5bi 183	. . . . . . 7 ⊢ ((G ‘w) = v → (w ∈ ℕ → v ≤ (H ‘1)))
32	31	com12 13	. . . . . 6 ⊢ (w ∈ ℕ → ((G ‘w) = v → v ≤ (H ‘1)))
33	32	r19.23aiv 1284	. . . . 5 ⊢ (∃w ∈ ℕ (G ‘w) = v → v ≤ (H ‘1))
34	18, 33	sylbi 174	. . . 4 ⊢ (v ∈ ran G → v ≤ (H ‘1))
35	34	rgen 1247	. . 3 ⊢ ∀v ∈ ran Gv ≤ (H ‘1)
36		breq2 2066	. . . . 5 ⊢ (w = (H ‘1) → (v ≤ w ↔ v ≤ (H ‘1)))
37	36	biraldv 1219	. . . 4 ⊢ (w = (H ‘1) → (∀v ∈ ran Gv ≤ w ↔ ∀v ∈ ran Gv ≤ (H ‘1)))
38	37	rcla4ev 1403	. . 3 ⊢ (((H ‘1) ∈ ℝ ∧ ∀v ∈ ran Gv ≤ (H ‘1)) → ∃w ∈ ℝ ∀v ∈ ran Gv ≤ w)
39	16, 35, 38	mp2an 520	. 2 ⊢ ∃w ∈ ℝ ∀v ∈ ran Gv ≤ w
40	8, 15, 39	3pm3.2i 603	1 ⊢ (ran G ⊆ ℝ ∧ ¬ ran G = ∅ ∧ ∃w ∈ ℝ ∀v ∈ ran Gv ≤ w)

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∖ cdif 1484   ∪ cun 1485   ⊆ wss 1487  ∅c0 1707  ifcif 1776  {csn 1808  ⟨cop 1810   class class class wbr 2054   × 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:  ruclem34 4918  ruclem35 4919

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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