Nuprl Lemma : qexp_step
∀[n:ℕ]. ∀[e:ℚ]. (e ↑ n + 1 = (e ↑ n * e) ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qmul: r * s,
rationals: ℚ,
nat: ℕ,
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
guard: {T}
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
rationals_wf,
exp_unroll_q,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
qmul_wf,
qexp_wf,
iff_weakening_equal,
add-subtract-cancel,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
imageMemberEquality,
baseClosed,
axiomEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[e:\mBbbQ{}]. (e \muparrow{} n + 1 = (e \muparrow{} n * e))
Date html generated:
2018_05_21-PM-11_59_16
Last ObjectModification:
2017_07_26-PM-06_48_37
Theory : rationals
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