Nuprl Lemma : add-div-when-divides2
∀a,b,x,y:ℤ. ∀c:ℤ-o. ((((a ÷ c) * x) + ((b ÷ c) * y)) = (((a * x) + (b * y)) ÷ c) ∈ ℤ) supposing ((c | a) and (c | b))
Proof
Definitions occuring in Statement :
divides: b | a,
int_nzero: ℤ-o,
uimplies: b supposing a,
all: ∀x:A. B[x],
divide: n ÷ m,
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
divides: b | a,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
prop: ℙ,
true: True,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
divides_wf,
int_nzero_wf,
istype-int,
divide_wfa,
subtype_base_sq,
int_subtype_base,
int_nzero_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
divide-exact,
equal_wf,
squash_wf,
true_wf,
istype-universe,
add_functionality_wrt_eq,
subtype_rel_self,
iff_weakening_equal,
div-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
Error :universeIsType,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
intEquality,
multiplyEquality,
because_Cache,
natural_numberEquality,
instantiate,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
voidElimination,
equalityTransitivity,
equalitySymmetry,
addEquality,
applyEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}a,b,x,y:\mBbbZ{}. \mforall{}c:\mBbbZ{}\msupminus{}\msupzero{}.
((((a \mdiv{} c) * x) + ((b \mdiv{} c) * y)) = (((a * x) + (b * y)) \mdiv{} c)) supposing ((c | a) and (c | b))
Date html generated:
2019_06_20-PM-02_20_38
Last ObjectModification:
2019_03_06-AM-11_06_00
Theory : num_thy_1
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