Nuprl Lemma : qexp-mul
∀[m,n:ℕ]. ∀[b:ℚ]. (b ↑ n * m = b ↑ n ↑ m ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
rationals: ℚ,
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
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,
so_lambda: λ2x.t[x],
decidable: Dec(P),
or: P ∨ Q,
true: True,
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
subtract: n - m
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,
rationals_wf,
nat_wf,
uall_wf,
squash_wf,
true_wf,
equal_wf,
qexp_wf,
decidable__le,
intformnot_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_term_value_mul_lemma,
le_wf,
exp_zero_q,
iff_weakening_equal,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
int-subtype-rationals,
mul-commutes,
zero-mul,
qexp-zero,
mul_bounds_1a,
exp_unroll_q,
qmul_wf,
qexp-add,
mul-distributes,
add-associates,
add-swap,
add-commutes,
add-mul-special,
zero-add
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,
functionEquality,
cumulativity,
universeEquality,
because_Cache,
dependent_set_memberEquality,
multiplyEquality,
unionElimination,
imageMemberEquality,
baseClosed,
productElimination,
minusEquality
Latex:
\mforall{}[m,n:\mBbbN{}]. \mforall{}[b:\mBbbQ{}]. (b \muparrow{} n * m = b \muparrow{} n \muparrow{} m)
Date html generated:
2018_05_22-AM-00_01_15
Last ObjectModification:
2017_07_26-PM-06_50_03
Theory : rationals
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