Nuprl Lemma : add-div-when-divides
∀a,b:ℤ. ∀c:ℤ-o. (((a ÷ c) + (b ÷ c)) = ((a + b) ÷ 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,
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,
top: Top,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
divides_wf,
int_nzero_wf,
istype-int,
divide_wfa,
mul-distributes,
istype-void,
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,
equalityTransitivity,
equalitySymmetry,
voidElimination,
addEquality,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
instantiate,
universeEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}a,b:\mBbbZ{}. \mforall{}c:\mBbbZ{}\msupminus{}\msupzero{}. (((a \mdiv{} c) + (b \mdiv{} c)) = ((a + b) \mdiv{} c)) supposing ((c | a) and (c | b))
Date html generated:
2019_06_20-PM-02_20_36
Last ObjectModification:
2019_03_06-AM-10_54_07
Theory : num_thy_1
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