Nuprl Lemma : rinv_functionality
∀[x,y:ℝ]. (rnonzero(x) ⇒ (x = y) ⇒ (rinv(x) = rinv(y)))
Proof
Definitions occuring in Statement :
rinv: rinv(x),
rnonzero: rnonzero(x),
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
rnonzero: rnonzero(x),
exists: ∃x:A. B[x],
rinv: rinv(x),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
has-value: (a)↓,
real: ℝ,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
sq_stable: SqStable(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
reg-seq-inv: reg-seq-inv(x),
req: x = y,
reg-seq-adjust: reg-seq-adjust(n;x)
Lemmas referenced :
rnonzero_functionality,
req-iff-bdd-diff,
rinv_wf,
subtype_rel_sets,
less_than_wf,
le_wf,
nat_plus_properties,
decidable__le,
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,
value-type-has-value,
int-value-type,
assert_of_lt_int,
absval_wf,
nat_wf,
assert_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
lt_int_wf,
mu-ge-property,
int_upper_wf,
mu-ge_wf,
nat_plus_wf,
int_upper_properties,
uall_wf,
int_seg_wf,
not_wf,
equal_wf,
req_wf,
rnonzero_wf,
req_witness,
rnonzero-lemma1,
sq_stable__le,
int_seg_properties,
set-value-type,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
reg-seq-inv_wf,
int_subtype_base,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
mul_bounds_1a,
nat_plus_subtype_nat,
reg-seq-adjust_wf,
regular-int-seq_wf,
ifthenelse_wf,
eq_int_eq_false,
iff_weakening_equal,
mul_nat_plus,
reg-seq-adjust-property,
nat_properties,
lelt_wf,
accelerate_wf,
bdd-diff_functionality,
accelerate-bdd-diff,
bdd-diff-iff-eventual,
imax_wf,
imax_nat_plus,
exists_wf,
all_wf,
subtract_wf,
less_than_transitivity1,
itermAdd_wf,
int_term_value_add_lemma,
rinv-functionality-lemma,
mul-associates,
mul-swap,
and_wf,
imax_ub,
top_wf,
not-equal-2,
le_antisymmetry_iff,
add-associates,
add-swap,
bfalse_wf,
less-iff-le,
add-zero,
add_nat_wf,
multiply_nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
productElimination,
isectElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
applyEquality,
intEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
setEquality,
applyLambdaEquality,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
callbyvalueReduce,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
imageElimination,
promote_hyp,
imageMemberEquality,
baseClosed,
equalityElimination,
instantiate,
cumulativity,
multiplyEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionEquality,
functionExtensionality,
addEquality,
addLevel,
hyp_replacement,
levelHypothesis,
inrFormation,
lessCases,
sqequalAxiom,
inlFormation
Latex:
\mforall{}[x,y:\mBbbR{}]. (rnonzero(x) {}\mRightarrow{} (x = y) {}\mRightarrow{} (rinv(x) = rinv(y)))
Date html generated:
2017_10_02-PM-07_16_55
Last ObjectModification:
2017_07_28-AM-07_21_06
Theory : reals
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