Nuprl Lemma : square-req-1-iff
∀x:ℝ. (x ≠ -(r1) ⇒ (x^2 = r1 ⇐⇒ x = r1))
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,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
or: P ∨ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
iff_wf,
all_wf,
le_wf,
false_wf,
rnexp_wf,
square-is-one,
real_wf,
rneq_wf,
rminus_wf,
int-to-real_wf,
req_wf,
or_wf,
req_inversion,
req_weakening,
rneq_functionality,
rneq_irreflexivity
Rules used in proof :
functionEquality,
lambdaEquality,
sqequalRule,
dependent_set_memberEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
impliesFunctionality,
allFunctionality,
addLevel,
inlFormation,
natural_numberEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
thin,
unionElimination,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut,
independent_isectElimination,
because_Cache,
voidElimination
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} -(r1) {}\mRightarrow{} (x\^{}2 = r1 \mLeftarrow{}{}\mRightarrow{} x = r1))
Date html generated:
2017_10_03-AM-08_50_39
Last ObjectModification:
2017_08_02-PM-03_11_59
Theory : reals
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