Theorem ruclem6 14964

Description: Lemma for ruc 14972. Domain and range 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 6192	. . . . . 6 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3		0z 11388	. . . . . . 7 ⊢ 0 ∈ ℤ
4		seq1 12814	. . . . . . 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 2644	. . . . 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 14963	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
11	6, 10	syl5eqr 2670	. . . 4 ⊢ (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12		0re 10040	. . . . 5 ⊢ 0 ∈ ℝ
13		1re 10039	. . . . 5 ⊢ 1 ∈ ℝ
14		opelxpi 5148	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
15	12, 13, 14	mp2an 708	. . . 4 ⊢ ⟨0, 1⟩ ∈ (ℝ × ℝ)
16	11, 15	syl6eqel 2709	. . 3 ⊢ (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17		1st2nd2 7205	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
18	17	ad2antrl 764	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
19	18	oveq1d 6665	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) = (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
20	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐹:ℕ⟶ℝ)
21	8	adantr 481	. . . . . 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 7198	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
23	22	ad2antrl 764	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st ‘𝑧) ∈ ℝ)
24		xp2nd 7199	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
25	24	ad2antrl 764	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd ‘𝑧) ∈ ℝ)
26		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27		eqid 2622	. . . . . 6 ⊢ (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
28		eqid 2622	. . . . . 6 ⊢ (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤))
29	20, 21, 23, 25, 26, 27, 28	ruclem1 14960	. . . . 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 1073	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
31	19, 30	eqeltrd 2701	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32		nn0uz 11722	. . 3 ⊢ ℕ0 = (ℤ≥‘0)
33		0zd 11389	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
34		0p1e1 11132	. . . . . . 7 ⊢ (0 + 1) = 1
35	34	fveq2i 6194	. . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1)
36		nnuz 11723	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
37	35, 36	eqtr4i 2647	. . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ
38	37	eleq2i 2693	. . . 4 ⊢ (𝑧 ∈ (ℤ≥‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
39	9	equncomi 3759	. . . . . . . 8 ⊢ 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
40	39	fveq1i 6192	. . . . . . 7 ⊢ (𝐶‘𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41		nnne0 11053	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0)
42	41	necomd 2849	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43		fvunsn 6445	. . . . . . . 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	syl5eq 2668	. . . . . 6 ⊢ (𝑧 ∈ ℕ → (𝐶‘𝑧) = (𝐹‘𝑧))
46	45	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) = (𝐹‘𝑧))
47	7	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ℝ)
48	46, 47	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) ∈ ℝ)
49	38, 48	sylan2b 492	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘(0 + 1))) → (𝐶‘𝑧) ∈ ℝ)
50	16, 31, 32, 33, 49	seqf2 12820	. 2 ⊢ (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
51	1	feq1i 6036	. 2 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
52	50, 51	sylibr 224	1 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ⦋csb 3533   ∪ cun 3572  ifcif 4086  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   / cdiv 10684  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  ruclem8  14966  ruclem9  14967  ruclem10  14968  ruclem11  14969  ruclem12  14970