Theorem sqrlem4 4734

Description: Lemma for square root theorem.

Hypotheses

Ref	Expression
sqrlem1.1	⊢ A ∈ ℝ
sqrlem1.2	⊢ 0 < A
sqrlem4.3	⊢ S = {x ∈ ℝ∣(0 ≤ x ∧ (x · x) ≤ A)}

Assertion

Ref	Expression
sqrlem4	⊢ (D ∈ S ↔ (D ∈ ℝ ∧ (0 ≤ D ∧ (D · D) ≤ A)))

Distinct variable group(s):   x,A   x,S   x,D

Proof of Theorem sqrlem4

Step	Hyp	Ref	Expression
1		sqrlem4.3	. . 3 ⊢ S = {x ∈ ℝ∣(0 ≤ x ∧ (x · x) ≤ A)}
2	1	eleq2i 1153	. 2 ⊢ (D ∈ S ↔ D ∈ {x ∈ ℝ∣(0 ≤ x ∧ (x · x) ≤ A)})
3		breq2 2066	. . . 4 ⊢ (x = D → (0 ≤ x ↔ 0 ≤ D))
4		opreq12 3008	. . . . . 6 ⊢ ((x = D ∧ x = D) → (x · x) = (D · D))
5	4	anidms 332	. . . . 5 ⊢ (x = D → (x · x) = (D · D))
6	5	breq1d 2071	. . . 4 ⊢ (x = D → ((x · x) ≤ A ↔ (D · D) ≤ A))
7	3, 6	anbi12d 476	. . 3 ⊢ (x = D → ((0 ≤ x ∧ (x · x) ≤ A) ↔ (0 ≤ D ∧ (D · D) ≤ A)))
8	7	elrab 1422	. 2 ⊢ (D ∈ {x ∈ ℝ∣(0 ≤ x ∧ (x · x) ≤ A)} ↔ (D ∈ ℝ ∧ (0 ≤ D ∧ (D · D) ≤ A)))
9	2, 8	bitr 151	1 ⊢ (D ∈ S ↔ (D ∈ ℝ ∧ (0 ≤ D ∧ (D · D) ≤ A)))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {crab 1204   class class class wbr 2054  (class class class)co 3001  ℝcr 4027  0cc0 4028   · cmulc 4032   < clt 4033   ≤ cle 4092

This theorem is referenced by:  sqrlem5 4735  sqrlem6 4736  sqrlem12 4742  sqrlem13 4743

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-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  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

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