Nuprl Lemma : square-req-iff
∀x,y:ℝ. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (x = y ⇐⇒ x^2 = y^2))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
assert: ↑b,
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
btrue: tt,
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A
Lemmas referenced :
req_functionality,
rabs_wf,
rabs_functionality,
req_weakening,
req_wf,
rnexp-req-iff-even,
less_than_wf,
rnexp_wf,
false_wf,
le_wf,
iff_wf,
rleq_wf,
int-to-real_wf,
real_wf,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
productElimination,
addLevel,
impliesFunctionality,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 \mleq{} x) {}\mRightarrow{} (r0 \mleq{} y) {}\mRightarrow{} (x = y \mLeftarrow{}{}\mRightarrow{} x\^{}2 = y\^{}2))
Date html generated:
2016_10_26-AM-09_10_38
Last ObjectModification:
2016_10_01-AM-11_42_42
Theory : reals
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