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Theorem ruclem6 16174

Description: Lemma for ruc 16182. 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 ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘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 6889	. . . . . 6 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3		0z 12565	. . . . . . 7 ⊢ 0 ∈ ℤ
4		seq1 13975	. . . . . . 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 2760	. . . . 5 ⊢ (𝐺‘0) = (𝐶‘0)
7		ruc.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
8		ruc.2	. . . . . 6 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
9		ruc.4	. . . . . 6 ⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
10	7, 8, 9, 1	ruclem4 16173	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
11	6, 10	eqtr3id 2786	. . . 4 ⊢ (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12		0re 11212	. . . . 5 ⊢ 0 ∈ ℝ
13		1re 11210	. . . . 5 ⊢ 1 ∈ ℝ
14		opelxpi 5712	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
15	12, 13, 14	mp2an 690	. . . 4 ⊢ ⟨0, 1⟩ ∈ (ℝ × ℝ)
16	11, 15	eqeltrdi 2841	. . 3 ⊢ (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17		1st2nd2 8010	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
18	17	ad2antrl 726	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
19	18	oveq1d 7420	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) = (⟨(1^st ‘𝑧), (2^nd ‘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 ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
22		xp1st 8003	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1^st ‘𝑧) ∈ ℝ)
23	22	ad2antrl 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1^st ‘𝑧) ∈ ℝ)
24		xp2nd 8004	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2^nd ‘𝑧) ∈ ℝ)
25	24	ad2antrl 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2^nd ‘𝑧) ∈ ℝ)
26		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27		eqid 2732	. . . . . 6 ⊢ (1^st ‘(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤)) = (1^st ‘(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤))
28		eqid 2732	. . . . . 6 ⊢ (2^nd ‘(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤)) = (2^nd ‘(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤))
29	20, 21, 23, 25, 26, 27, 28	ruclem1 16170	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → ((⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ) ∧ (1^st ‘(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤)) = if((((1^st ‘𝑧) + (2^nd ‘𝑧)) / 2) < 𝑤, (1^st ‘𝑧), (((((1^st ‘𝑧) + (2^nd ‘𝑧)) / 2) + (2^nd ‘𝑧)) / 2)) ∧ (2^nd ‘(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤)) = if((((1^st ‘𝑧) + (2^nd ‘𝑧)) / 2) < 𝑤, (((1^st ‘𝑧) + (2^nd ‘𝑧)) / 2), (2^nd ‘𝑧))))
30	29	simp1d 1142	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
31	19, 30	eqeltrd 2833	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32		nn0uz 12860	. . 3 ⊢ ℕ₀ = (ℤ_≥‘0)
33		0zd 12566	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
34		0p1e1 12330	. . . . . . 7 ⊢ (0 + 1) = 1
35	34	fveq2i 6891	. . . . . 6 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
36		nnuz 12861	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
37	35, 36	eqtr4i 2763	. . . . 5 ⊢ (ℤ_≥‘(0 + 1)) = ℕ
38	37	eleq2i 2825	. . . 4 ⊢ (𝑧 ∈ (ℤ_≥‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
39	9	equncomi 4154	. . . . . . . 8 ⊢ 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
40	39	fveq1i 6889	. . . . . . 7 ⊢ (𝐶‘𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41		nnne0 12242	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0)
42	41	necomd 2996	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43		fvunsn 7173	. . . . . . . 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 2784	. . . . . 6 ⊢ (𝑧 ∈ ℕ → (𝐶‘𝑧) = (𝐹‘𝑧))
46	45	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐶‘𝑧) = (𝐹‘𝑧))
47	7	ffvelcdmda 7083	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ℝ)
48	46, 47	eqeltrd 2833	. . . 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 13983	. 2 ⊢ (𝜑 → seq0(𝐷, 𝐶):ℕ₀⟶(ℝ × ℝ))
51	1	feq1i 6705	. 2 ⊢ (𝐺:ℕ₀⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ₀⟶(ℝ × ℝ))
52	50, 51	sylibr 233	1 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⦋csb 3892 ∪ cun 3945 ifcif 4527 {csn 4627 ⟨cop 4633 class class class wbr 5147 × cxp 5673 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7969 2^nd c2nd 7970 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 seqcseq 13962

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-seq 13963

This theorem is referenced by: ruclem8 16176 ruclem9 16177 ruclem10 16178 ruclem11 16179 ruclem12 16180

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