Nuprl Lemma : rnonzero_functionality
∀x,y:ℝ. ((x = y) ⇒ (rnonzero(x) ⇐⇒ rnonzero(y)))
Proof
Definitions occuring in Statement :
rnonzero: rnonzero(x),
req: x = y,
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
real: ℝ,
rnonzero: rnonzero(x),
exists: ∃x:A. B[x],
uimplies: b supposing a,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than': less_than'(a;b),
true: True,
req: x = y,
le: A ≤ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rnonzero_wf,
req_wf,
real_wf,
rnonzero-lemma1,
nat_plus_properties,
decidable__le,
absval_wf,
less_than_wf,
nat_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
mul_nat_plus,
itermMultiply_wf,
int_term_value_mul_lemma,
subtract_wf,
multiply-is-int-iff,
false_wf,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
and_wf,
equal_wf,
le_wf,
le_functionality,
le_weakening,
int-triangle-inequality,
mul_cancel_in_le,
decidable__lt,
add-is-int-iff,
req_inversion
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
applyEquality,
dependent_set_memberEquality,
lambdaEquality,
sqequalRule,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
because_Cache,
independent_functionElimination,
multiplyEquality,
addEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
setEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (rnonzero(x) \mLeftarrow{}{}\mRightarrow{} rnonzero(y)))
Date html generated:
2016_10_26-AM-09_04_12
Last ObjectModification:
2016_07_12-AM-08_14_38
Theory : reals
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