Theorem ruclem6 16267

Description: Lemma for ruc 16275. 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 6907	. . . . . 6 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3		0z 12621	. . . . . . 7 ⊢ 0 ∈ ℤ
4		seq1 14051	. . . . . . 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 2762	. . . . 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 16266	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
11	6, 10	eqtr3id 2788	. . . 4 ⊢ (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12		0re 11260	. . . . 5 ⊢ 0 ∈ ℝ
13		1re 11258	. . . . 5 ⊢ 1 ∈ ℝ
14		opelxpi 5725	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
15	12, 13, 14	mp2an 692	. . . 4 ⊢ ⟨0, 1⟩ ∈ (ℝ × ℝ)
16	11, 15	eqeltrdi 2846	. . 3 ⊢ (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17		1st2nd2 8051	. . . . . 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 7445	. . . 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 8044	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
23	22	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st ‘𝑧) ∈ ℝ)
24		xp2nd 8045	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
25	24	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd ‘𝑧) ∈ ℝ)
26		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27		eqid 2734	. . . . . 6 ⊢ (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
28		eqid 2734	. . . . . 6 ⊢ (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
29	20, 21, 23, 25, 26, 27, 28	ruclem1 16263	. . . . 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 1141	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
31	19, 30	eqeltrd 2838	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32		nn0uz 12917	. . 3 ⊢ ℕ0 = (ℤ≥‘0)
33		0zd 12622	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
34		0p1e1 12385	. . . . . . 7 ⊢ (0 + 1) = 1
35	34	fveq2i 6909	. . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1)
36		nnuz 12918	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
37	35, 36	eqtr4i 2765	. . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ
38	37	eleq2i 2830	. . . 4 ⊢ (𝑧 ∈ (ℤ≥‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
39	9	equncomi 4169	. . . . . . . 8 ⊢ 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
40	39	fveq1i 6907	. . . . . . 7 ⊢ (𝐶‘𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41		nnne0 12297	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0)
42	41	necomd 2993	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43		fvunsn 7198	. . . . . . . 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 2786	. . . . . 6 ⊢ (𝑧 ∈ ℕ → (𝐶‘𝑧) = (𝐹‘𝑧))
46	45	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) = (𝐹‘𝑧))
47	7	ffvelcdmda 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ℝ)
48	46, 47	eqeltrd 2838	. . . 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 14058	. 2 ⊢ (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
51	1	feq1i 6727	. 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 1536   ∈ wcel 2105   ≠ wne 2937  ⦋csb 3907   ∪ cun 3960  ifcif 4530  {csn 4630  ⟨cop 4636   class class class wbr 5147   × cxp 5686  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8010  2^nd c2nd 8011  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292   / cdiv 11917  ℕcn 12263  2c2 12318  ℕ₀cn0 12523  ℤcz 12610  ℤ_≥cuz 12875  seqcseq 14038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-seq 14039

This theorem is referenced by:  ruclem8  16269  ruclem9  16270  ruclem10  16271  ruclem11  16272  ruclem12  16273