Theorem ruclem12 4896

Description: Lemma for ruc 4924. A helper lemma that changes bound variables.

Hypothesis

Ref	Expression
ruclem12.2	⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}

Assertion

Ref	Expression
ruclem12	⊢ D = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))}

Distinct variable group(s):   x,y,z,w,v,u

Proof of Theorem ruclem12

Step	Hyp	Ref	Expression
1		ruclem12.2	. 2 ⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}
2		eleq1 1149	. . . . . 6 ⊢ (x = w → (x ∈ (ℝ × ℝ) ↔ w ∈ (ℝ × ℝ)))
3	2	anbi1d 469	. . . . 5 ⊢ (x = w → ((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ↔ (w ∈ (ℝ × ℝ) ∧ y ∈ ℝ)))
4		eleq1 1149	. . . . . 6 ⊢ (y = v → (y ∈ ℝ ↔ v ∈ ℝ))
5	4	anbi2d 468	. . . . 5 ⊢ (y = v → ((w ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ↔ (w ∈ (ℝ × ℝ) ∧ v ∈ ℝ)))
6	3, 5	sylan9bb 418	. . . 4 ⊢ ((x = w ∧ y = v) → ((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ↔ (w ∈ (ℝ × ℝ) ∧ v ∈ ℝ)))
7		ruclem4 4888	. . . . 5 ⊢ ((x = w ∧ y = v) → if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩) = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))
8	7	cleq2d 1112	. . . 4 ⊢ ((x = w ∧ y = v) → (z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩) ↔ z = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩)))
9	6, 8	anbi12d 476	. . 3 ⊢ ((x = w ∧ y = v) → (((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩)) ↔ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ z = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))))
10	9	cbvoprab12v 3029	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))} = {⟨⟨w, v⟩, z⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ z = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))}
11		cleq1 1107	. . . 4 ⊢ (z = u → (z = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ↔ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩)))
12	11	anbi2d 468	. . 3 ⊢ (z = u → (((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ z = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩)) ↔ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))))
13	12	cbvoprab3v 3030	. 2 ⊢ {⟨⟨w, v⟩, z⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ z = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))} = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))}
14	1, 10, 13	3eqtr 1123	1 ⊢ D = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))}

Syntax hints:   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ifcif 1776  ⟨cop 1810   class class class wbr 2054   × cxp 2408   ‘cfv 2422  (class class class)co 3001  {copab2 3002  1^st c1st 3085  2^nd c2nd 3086  ℝcr 4027   + caddc 4031   · cmulc 4032   < clt 4033   / cdiv 4091  2c2 4454  3c3 4455

This theorem is referenced by:  ruclem13 4897  ruclem15 4899

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-oprab 3004

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