Nuprl Lemma : exp_functionality_wrt_eqmod
∀m,i,j:ℤ. ((i ≡ j mod m) ⇒ (∀n:ℕ. (i^n ≡ j^n mod m)))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
eqmod_wf,
exp_wf2,
subtract_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
primrec-wf2,
istype-nat,
exp0_lemma,
eqmod_weakening,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
istype-void,
squash_wf,
true_wf,
exp_add,
exp1,
subtype_rel_self,
iff_weakening_equal,
eqmod_refl,
eqmod_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
rename,
setElimination,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
voidElimination,
setIsType,
because_Cache,
inhabitedIsType,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
imageElimination,
multiplyEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination
Latex:
\mforall{}m,i,j:\mBbbZ{}. ((i \mequiv{} j mod m) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. (i\^{}n \mequiv{} j\^{}n mod m)))
Date html generated:
2020_05_19-PM-10_01_54
Last ObjectModification:
2020_01_01-AM-10_11_02
Theory : num_thy_1
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