Nuprl Lemma : square-is-one
∀x:ℝ. (x^2 = r1 ⇐⇒ (x = r1) ∨ (x = -(r1)))
Proof
Definitions occuring in Statement :
rnexp: x^k1,
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
not: ¬A,
false: False,
le: A ≤ B,
nat: ℕ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a
Lemmas referenced :
iff_wf,
all_wf,
le_wf,
false_wf,
rnexp_wf,
rless_wf,
rless-int,
squares-req,
real_wf,
rminus_wf,
int-to-real_wf,
req_wf,
or_wf,
rnexp-one,
req_weakening,
req_functionality
Rules used in proof :
cut,
lambdaEquality,
dependent_set_memberEquality,
baseClosed,
imageMemberEquality,
inrFormation,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
impliesFunctionality,
productElimination,
allFunctionality,
addLevel,
because_Cache,
natural_numberEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
promote_hyp,
independent_isectElimination
Latex:
\mforall{}x:\mBbbR{}. (x\^{}2 = r1 \mLeftarrow{}{}\mRightarrow{} (x = r1) \mvee{} (x = -(r1)))
Date html generated:
2017_10_03-AM-08_50_30
Last ObjectModification:
2017_08_02-PM-03_10_31
Theory : reals
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