Theorem ruclem6 16403

Description: Lemma for ruc 16411. 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 6886	. . . . . 6 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3		0z 12704	. . . . . . 7 ⊢ 0 ∈ ℤ
4		seq1 14157	. . . . . . 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 2784	. . . . 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 16402	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
11	6, 10	eqtr3id 2810	. . . 4 ⊢ (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12		0re 11310	. . . . 5 ⊢ 0 ∈ ℝ
13		1re 11308	. . . . 5 ⊢ 1 ∈ ℝ
14		opelxpi 5688	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
15	12, 13, 14	mp2an 705	. . . 4 ⊢ ⟨0, 1⟩ ∈ (ℝ × ℝ)
16	11, 15	eqeltrdi 2869	. . 3 ⊢ (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17		1st2nd2 8040	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
18	17	ad2antrl 741	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
19	18	oveq1d 7435	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) = (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
20	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐹:ℕ⟶ℝ)
21	8	adantr 486	. . . . . 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 8033	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
23	22	ad2antrl 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st ‘𝑧) ∈ ℝ)
24		xp2nd 8034	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
25	24	ad2antrl 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd ‘𝑧) ∈ ℝ)
26		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27		eqid 2761	. . . . . 6 ⊢ (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
28		eqid 2761	. . . . . 6 ⊢ (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
29	20, 21, 23, 25, 26, 27, 28	ruclem1 16399	. . . . 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 1160	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
31	19, 30	eqeltrd 2861	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32		nn0uz 13003	. . 3 ⊢ ℕ0 = (ℤ≥‘0)
33		0zd 12705	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
34		0p1e1 12463	. . . . . . 7 ⊢ (0 + 1) = 1
35	34	fveq2i 6888	. . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1)
36		nnuz 13004	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
37	35, 36	eqtr4i 2787	. . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ
38	37	eleq2i 2853	. . . 4 ⊢ (𝑧 ∈ (ℤ≥‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
39	9	equncomi 4107	. . . . . . . 8 ⊢ 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
40	39	fveq1i 6886	. . . . . . 7 ⊢ (𝐶‘𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41		nnne0 12372	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0)
42	41	necomd 3011	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43		fvunsn 7184	. . . . . . . 8 ⊢ (0 ≠ 𝑧 → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹‘𝑧))
44	42, 43	syl 18	. . . . . . 7 ⊢ (𝑧 ∈ ℕ → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹‘𝑧))
45	40, 44	eqtrid 2808	. . . . . 6 ⊢ (𝑧 ∈ ℕ → (𝐶‘𝑧) = (𝐹‘𝑧))
46	45	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) = (𝐹‘𝑧))
47	7	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ℝ)
48	46, 47	eqeltrd 2861	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) ∈ ℝ)
49	38, 48	sylan2b 606	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘(0 + 1))) → (𝐶‘𝑧) ∈ ℝ)
50	16, 31, 32, 33, 49	seqf2 14164	. 2 ⊢ (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
51	1	feq1i 6700	. 2 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
52	50, 51	sylibr 237	1 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ⦋csb 3847   ∪ cun 3897  ifcif 4482  {csn 4584  ⟨cop 4590   class class class wbr 5103   × cxp 5649  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   < clt 11343   / cdiv 11973  ℕcn 12335  2c2 12397  ℕ₀cn0 12606  ℤcz 12693  ℤ_≥cuz 12965  seqcseq 14144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-seq 14145

This theorem is used by:  ruclem8  16405  ruclem9  16406  ruclem10  16407  ruclem11  16408  ruclem12  16409