Nuprl Lemma : inverse-rpower
∀[x:ℝ]. ∀[n:ℕ]. ((r1/x^n) = (r1/x)^n) supposing x ≠ r0
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
rat_term_to_real: rat_term_to_real(f;t),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
pi1: fst(t),
true: True,
rtermDivide: num "/" denom,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
pi2: snd(t),
decidable: Dec(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
rtermMultiply: left "*" right,
rtermVar: rtermVar(var)
Lemmas referenced :
rpower-nonzero,
req_witness,
rdiv_wf,
int-to-real_wf,
rnexp_wf,
istype-nat,
rneq_wf,
real_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
rnexp_zero_lemma,
assert-rat-term-eq2,
rtermDivide_wf,
rtermConstant_wf,
rless-int,
rless_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract-1-ge-0,
ifthenelse_wf,
eq_int_wf,
rmul_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rneq_functionality,
rnexp-req,
req_weakening,
req_functionality,
rdiv_functionality,
rtermMultiply_wf,
rtermVar_wf,
rmul_functionality,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
inhabitedIsType,
isectElimination,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
universeIsType,
setElimination,
rename,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
inrFormation_alt,
productElimination,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[n:\mBbbN{}]. ((r1/x\^{}n) = (r1/x)\^{}n) supposing x \mneq{} r0
Date html generated:
2019_10_29-AM-09_58_04
Last ObjectModification:
2019_04_01-PM-11_19_47
Theory : reals
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