Nuprl Lemma : iseg_product-property
∀i,j:ℤ. ∀k:ℕ. (k | iseg_product(i;j)) supposing ((k ≤ j) and (i ≤ k))
Proof
Definitions occuring in Statement :
iseg_product: iseg_product(i;j),
divides: b | a,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
iseg_product: iseg_product(i;j),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
nat_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
subtract_wf,
divides-combinations,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}i,j:\mBbbZ{}. \mforall{}k:\mBbbN{}. (k | iseg\_product(i;j)) supposing ((k \mleq{} j) and (i \mleq{} k))
Date html generated:
2016_05_15-PM-06_01_27
Last ObjectModification:
2016_01_16-PM-00_40_23
Theory : general
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