Theorem ruclem6 16144

Description: Lemma for ruc 16152. Domain and codomain of the interval sequence. (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 ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)))
ruc.4	⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5	⊢ 𝐺 = seq0(𝐷, 𝐶)

Assertion

Ref	Expression
ruclem6	⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))

Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑚,𝐺,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem6

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ruc.5	. . . . . . 7 ⊢ 𝐺 = seq0(𝐷, 𝐶)
2	1	fveq1i 6823	. . . . . 6 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3		0z 12479	. . . . . . 7 ⊢ 0 ∈ ℤ
4		seq1 13921	. . . . . . 7 ⊢ (0 ∈ ℤ → (seq0(𝐷, 𝐶)‘0) = (𝐶‘0))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (seq0(𝐷, 𝐶)‘0) = (𝐶‘0)
6	2, 5	eqtri 2754	. . . . 5 ⊢ (𝐺‘0) = (𝐶‘0)
7		ruc.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
8		ruc.2	. . . . . 6 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)))
9		ruc.4	. . . . . 6 ⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
10	7, 8, 9, 1	ruclem4 16143	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
11	6, 10	eqtr3id 2780	. . . 4 ⊢ (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12		0re 11114	. . . . 5 ⊢ 0 ∈ ℝ
13		1re 11112	. . . . 5 ⊢ 1 ∈ ℝ
14		opelxpi 5651	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
15	12, 13, 14	mp2an 692	. . . 4 ⊢ ⟨0, 1⟩ ∈ (ℝ × ℝ)
16	11, 15	eqeltrdi 2839	. . 3 ⊢ (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17		1st2nd2 7960	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
18	17	ad2antrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
19	18	oveq1d 7361	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) = (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
20	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐹:ℕ⟶ℝ)
21	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)))
22		xp1st 7953	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
23	22	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st ‘𝑧) ∈ ℝ)
24		xp2nd 7954	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
25	24	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd ‘𝑧) ∈ ℝ)
26		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27		eqid 2731	. . . . . 6 ⊢ (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
28		eqid 2731	. . . . . 6 ⊢ (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
29	20, 21, 23, 25, 26, 27, 28	ruclem1 16140	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → ((⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ) ∧ (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = if((((1st ‘𝑧) + (2nd ‘𝑧)) / 2) < 𝑤, (1st ‘𝑧), (((((1st ‘𝑧) + (2nd ‘𝑧)) / 2) + (2nd ‘𝑧)) / 2)) ∧ (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = if((((1st ‘𝑧) + (2nd ‘𝑧)) / 2) < 𝑤, (((1st ‘𝑧) + (2nd ‘𝑧)) / 2), (2nd ‘𝑧))))
30	29	simp1d 1142	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
31	19, 30	eqeltrd 2831	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32		nn0uz 12774	. . 3 ⊢ ℕ0 = (ℤ≥‘0)
33		0zd 12480	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
34		0p1e1 12242	. . . . . . 7 ⊢ (0 + 1) = 1
35	34	fveq2i 6825	. . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1)
36		nnuz 12775	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
37	35, 36	eqtr4i 2757	. . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ
38	37	eleq2i 2823	. . . 4 ⊢ (𝑧 ∈ (ℤ≥‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
39	9	equncomi 4107	. . . . . . . 8 ⊢ 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
40	39	fveq1i 6823	. . . . . . 7 ⊢ (𝐶‘𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41		nnne0 12159	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0)
42	41	necomd 2983	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43		fvunsn 7113	. . . . . . . 8 ⊢ (0 ≠ 𝑧 → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹‘𝑧))
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ ℕ → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹‘𝑧))
45	40, 44	eqtrid 2778	. . . . . 6 ⊢ (𝑧 ∈ ℕ → (𝐶‘𝑧) = (𝐹‘𝑧))
46	45	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) = (𝐹‘𝑧))
47	7	ffvelcdmda 7017	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ℝ)
48	46, 47	eqeltrd 2831	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) ∈ ℝ)
49	38, 48	sylan2b 594	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘(0 + 1))) → (𝐶‘𝑧) ∈ ℝ)
50	16, 31, 32, 33, 49	seqf2 13928	. 2 ⊢ (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
51	1	feq1i 6642	. 2 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
52	50, 51	sylibr 234	1 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ⦋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   / cdiv 11774  ℕcn 12125  2c2 12180  ℕ₀cn0 12381  ℤcz 12468  ℤ_≥cuz 12732  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:  ruclem8  16146  ruclem9  16147  ruclem10  16148  ruclem11  16149  ruclem12  16150