Nuprl Lemma : squares-req
∀x,y:ℝ. (y ≠ r0 ⇒ (x^2 = y^2 ⇐⇒ (x = y) ∨ (x = -(y))))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rnexp: x^k1,
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
nat_plus: ℕ+,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
btrue: tt,
true: True,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
less_than: a < b,
squash: ↓T,
isOdd: isOdd(n),
subtype_rel: A ⊆r B,
stable: Stable{P}
Latex:
\mforall{}x,y:\mBbbR{}. (y \mneq{} r0 {}\mRightarrow{} (x\^{}2 = y\^{}2 \mLeftarrow{}{}\mRightarrow{} (x = y) \mvee{} (x = -(y))))
Date html generated:
2020_05_20-AM-11_02_39
Last ObjectModification:
2020_01_08-AM-10_38_07
Theory : reals
Home
Index