Nuprl Lemma : div-mul-cancel
∀[a:ℤ]. ∀[n,m:ℤ-o]. ((a * n) ÷ m * n ~ a ÷ m)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
multiply: n * m,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_nzero: ℤ-o,
top: Top,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
int_nzero_wf,
equal_wf,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_nzero_properties,
div-cancel,
div_div,
mul-commutes,
int_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
multiplyEquality,
divideEquality,
lambdaFormation,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n,m:\mBbbZ{}\msupminus{}\msupzero{}]. ((a * n) \mdiv{} m * n \msim{} a \mdiv{} m)
Date html generated:
2016_05_14-AM-07_24_50
Last ObjectModification:
2016_01_14-PM-10_01_18
Theory : int_2
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