Nuprl Lemma : div-cancel-1
∀a:ℤ. ∀b:ℤ-o. (((a * b) ÷ b) = a ∈ ℤ)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
all: ∀x:A. B[x],
divide: n ÷ m,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
exists: ∃x:A. B[x],
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
implies: P ⇒ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
absval: |i|,
nequal: a ≠ b ∈ T
Lemmas referenced :
div_unique3,
add-zero,
istype-false,
istype-le,
istype-less_than,
absval_wf,
set_subtype_base,
nequal_wf,
int_subtype_base,
int_nzero_wf,
istype-int,
minus-zero,
absval-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
multiplyEquality,
hypothesisEquality,
setElimination,
rename,
hypothesis,
isectElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation_alt,
natural_numberEquality,
independent_pairFormation,
imageElimination,
sqequalRule,
because_Cache,
voidElimination,
productIsType,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
intEquality,
sqequalBase,
functionIsType,
universeIsType,
independent_functionElimination
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}b:\mBbbZ{}\msupminus{}\msupzero{}. (((a * b) \mdiv{} b) = a)
Date html generated:
2020_05_19-PM-09_35_31
Last ObjectModification:
2019_10_16-PM-02_00_08
Theory : arithmetic
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