Nuprl Lemma : rnonneg_functionality
∀x,y:ℝ. rnonneg(x) ⇐⇒ rnonneg(y) supposing x = y
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
req: x = y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
req: x = y,
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
real: ℝ,
subtype_rel: A ⊆r B,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rnonneg: rnonneg(x)
Lemmas referenced :
false_wf,
le_wf,
all_wf,
nat_plus_wf,
absval_wf,
subtract_wf,
nat_wf,
rnonneg2_wf,
rnonneg2_functionality,
iff_wf,
rnonneg-iff,
rnonneg_wf,
less_than'_wf,
req_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
independent_pairEquality,
voidElimination,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x,y:\mBbbR{}. rnonneg(x) \mLeftarrow{}{}\mRightarrow{} rnonneg(y) supposing x = y
Date html generated:
2016_05_18-AM-07_01_44
Last ObjectModification:
2015_12_28-AM-00_34_00
Theory : reals
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