Nuprl Lemma : rinv_wf
∀[x:ℝ]. (rnonzero(x) ⇒ (rinv(x) ∈ ℝ))
Proof
Definitions occuring in Statement :
rinv: rinv(x),
rnonzero: rnonzero(x),
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
rinv: rinv(x),
rnonzero: rnonzero(x),
real: ℝ,
prop: ℙ,
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
squash: ↓T,
nat: ℕ,
true: True,
less_than': less_than'(a;b),
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
and: P ∧ Q,
le: A ≤ B,
int_upper: {i...},
nat_plus: ℕ+,
has-value: (a)↓,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
rnonzero_wf,
real_wf,
bool_wf,
assert_wf,
int_upper_wf,
exists_wf,
squash_wf,
mu-ge_wf,
nat_wf,
absval_wf,
lt_int_wf,
less_than_wf,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
false_wf,
decidable__lt,
int-value-type,
value-type-has-value,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
le_wf,
subtype_rel_sets,
assert_of_lt_int,
subtype_base_sq,
set_subtype_base,
istype-int,
int_subtype_base,
subtype_rel_sets_simple,
istype-void,
istype-less_than,
istype-assert,
istype-false,
mu-ge-property,
istype-int_upper,
iff_weakening_uiff,
int_seg_wf,
rnonzero-lemma1,
int_upper_properties,
set-value-type,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
accelerate_wf,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_upper,
nequal-le-implies,
istype-le,
multiply_nat_plus,
add_nat_plus,
mul-non-neg1,
intformeq_wf,
int_formula_prop_eq_lemma,
reg-seq-inv_wf,
nat_plus_wf,
subtype_rel_self,
regular-int-seq_wf,
nat_plus_properties,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
reg-seq-adjust_wf,
mul_nat_plus,
mul-associates,
mul-swap,
reg-seq-adjust-property,
mul_preserves_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
hypothesis,
universeIsType,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
baseClosed,
imageMemberEquality,
intEquality,
functionEquality,
independent_isectElimination,
imageElimination,
lambdaEquality,
applyEquality,
natural_numberEquality,
because_Cache,
voidEquality,
isect_memberEquality,
independent_functionElimination,
voidElimination,
lambdaFormation,
independent_pairFormation,
unionElimination,
productElimination,
dependent_set_memberEquality,
callbyvalueReduce,
int_eqEquality,
approximateComputation,
setEquality,
dependent_pairFormation,
instantiate,
cumulativity,
closedConclusion,
dependent_pairFormation_alt,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
applyLambdaEquality,
equalityElimination,
equalityIsType4,
baseApply,
promote_hyp,
hypothesis_subsumption,
multiplyEquality,
equalityIsType1,
productIsType,
isectIsType,
functionIsType,
pointwiseFunctionality,
addEquality
Latex:
\mforall{}[x:\mBbbR{}]. (rnonzero(x) {}\mRightarrow{} (rinv(x) \mmember{} \mBbbR{}))
Date html generated:
2019_10_16-PM-03_07_30
Last ObjectModification:
2018_11_13-AM-10_32_46
Theory : reals
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