Nuprl Lemma : rmul-rinv1
∀[x:ℝ]. (rnonzero(x) ⇒ ((x * rinv(x)) = r1))
Proof
Definitions occuring in Statement :
rinv: rinv(x),
rnonzero: rnonzero(x),
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
real: ℝ,
subtype_rel: A ⊆r B,
rinv: rinv(x),
rnonzero: rnonzero(x),
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
has-value: (a)↓,
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nat: ℕ,
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,
bdd-diff: bdd-diff(f;g),
ge: i ≥ j ,
int-to-real: r(n),
reg-seq-inv: reg-seq-inv(x),
reg-seq-mul: reg-seq-mul(x;y),
nequal: a ≠ b ∈ T ,
absval: |i|,
int_nzero: ℤ-o,
subtract: n - m,
sq_stable: SqStable(P),
reg-seq-adjust: reg-seq-adjust(n;x)
Lemmas referenced :
req-iff-bdd-diff,
rmul_wf,
rinv_wf,
int-to-real_wf,
rnonzero_wf,
req_witness,
real_wf,
reg-seq-mul_wf,
subtype_base_sq,
int_upper_wf,
set_subtype_base,
le_wf,
int_subtype_base,
mu-ge_wf,
value-type-has-value,
int-value-type,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
lt_int_wf,
absval_wf,
subtype_rel_sets,
decidable__le,
full-omega-unsat,
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,
assert_of_lt_int,
nat_wf,
assert_wf,
bdd-diff_functionality,
rmul-bdd-diff-reg-seq-mul,
bdd-diff_weakening,
mu-ge-property,
int_seg_wf,
rnonzero-lemma1,
int_upper_properties,
set-value-type,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_upper,
nequal-le-implies,
uall_wf,
not_wf,
reg-seq-inv_wf,
nat_plus_properties,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
nat_plus_wf,
subtype_rel_self,
regular-int-seq_wf,
bdd-diff_wf,
squash_wf,
true_wf,
reg-seq-mul-comm,
accelerate_wf,
iff_weakening_equal,
reg-seq-mul_functionality_wrt_bdd-diff,
accelerate-bdd-diff,
canonical-bound_wf,
all_wf,
add_nat_wf,
subtype_rel_set,
upper_subtype_nat,
nat_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
subtract_wf,
equal-wf-T-base,
mul_cancel_in_le,
equal-wf-base,
absval_nat_plus,
absval_mul,
div_rem_sum2,
nequal_wf,
left_mul_subtract_distrib,
mul-commutes,
add-associates,
mul-associates,
mul-distributes,
minus-one-mul,
mul-swap,
one-mul,
add-swap,
mul-distributes-right,
zero-mul,
le_functionality,
int-triangle-inequality,
le_weakening,
add_functionality_wrt_eq,
absval_sym,
rem_bounds_absval,
set_wf,
sq_stable__less_than,
sq_stable__le,
absval_pos,
mul_bounds_1a,
reg-seq-adjust_wf,
lelt_wf,
mul_nat_plus,
reg-seq-adjust-property,
mul_preserves_le,
and_wf,
bdd-diff-iff-eventual,
exists_wf,
less_than_transitivity1,
top_wf,
not-equal-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
natural_numberEquality,
productElimination,
independent_isectElimination,
setElimination,
rename,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
instantiate,
cumulativity,
intEquality,
imageElimination,
callbyvalueReduce,
dependent_set_memberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
dependent_pairFormation,
setEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
applyLambdaEquality,
equalityElimination,
hypothesis_subsumption,
productEquality,
multiplyEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionEquality,
universeEquality,
addEquality,
minusEquality,
divideEquality,
remainderEquality,
hyp_replacement,
lessCases,
axiomSqEquality
Latex:
\mforall{}[x:\mBbbR{}]. (rnonzero(x) {}\mRightarrow{} ((x * rinv(x)) = r1))
Date html generated:
2019_10_16-PM-03_07_40
Last ObjectModification:
2018_08_27-PM-00_08_07
Theory : reals
Home
Index