Nuprl Lemma : rpositive_functionality
∀x,y:ℝ. rpositive(x) ⇐⇒ rpositive(y) supposing x = y
Proof
Definitions occuring in Statement :
rpositive: rpositive(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,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
real: ℝ,
subtype_rel: A ⊆r B,
so_apply: x[s],
guard: {T},
req: x = y
Lemmas referenced :
req_witness,
rpositive-iff,
false_wf,
le_wf,
nat_plus_wf,
all_wf,
absval_wf,
subtract_wf,
rpositive_wf,
req_wf,
real_wf,
req_inversion,
rpositive2_functionality,
bdd-diff_inversion
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
because_Cache,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaEquality,
applyEquality,
setElimination,
dependent_functionElimination,
independent_isectElimination
Latex:
\mforall{}x,y:\mBbbR{}. rpositive(x) \mLeftarrow{}{}\mRightarrow{} rpositive(y) supposing x = y
Date html generated:
2016_05_18-AM-07_00_52
Last ObjectModification:
2015_12_28-AM-00_33_30
Theory : reals
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