Nuprl Lemma : qexp-qdiv
∀[a,b:ℚ]. ∀[n:ℕ]. ((a/b) ↑ n = (a ↑ n/b ↑ n) ∈ ℚ) supposing ¬(b = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qdiv: (r/s),
rationals: ℚ,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
eq_int: (i =z j),
bfalse: ff,
assert: ↑b,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
cand: A c∧ B
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_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,
ge_wf,
less_than_wf,
equal_wf,
squash_wf,
true_wf,
exp_zero_q,
qdiv_wf,
not_wf,
equal-wf-T-base,
rationals_wf,
qexp-non-zero,
false_wf,
le_wf,
iff_weakening_equal,
int-subtype-rationals,
qdiv-self,
assert-qeq,
equal-wf-base,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
qmul-not-zero,
qexp_wf,
qmul_wf,
exp_unroll_q,
qmul-qdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
baseClosed,
dependent_set_memberEquality,
imageMemberEquality,
productElimination,
unionElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. \mforall{}[n:\mBbbN{}]. ((a/b) \muparrow{} n = (a \muparrow{} n/b \muparrow{} n)) supposing \mneg{}(b = 0)
Date html generated:
2018_05_22-AM-00_01_21
Last ObjectModification:
2017_07_26-PM-06_50_08
Theory : rationals
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