Nuprl Lemma : multiply_functionality_wrt_le
∀[i1,i2:ℕ]. ∀[j1,j2:ℤ]. ((i1 * i2) ≤ (j1 * j2)) supposing ((i2 ≤ j2) and (i1 ≤ j1))
Proof
Definitions occuring in Statement :
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
prop: ℙ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
int_term_value_mul_lemma,
itermMultiply_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
mul_preserves_le,
nat_wf,
le_wf,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
multiplyEquality,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
intEquality,
voidElimination,
independent_isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[i1,i2:\mBbbN{}]. \mforall{}[j1,j2:\mBbbZ{}]. ((i1 * i2) \mleq{} (j1 * j2)) supposing ((i2 \mleq{} j2) and (i1 \mleq{} j1))
Date html generated:
2016_05_14-AM-07_20_33
Last ObjectModification:
2016_01_07-PM-03_59_57
Theory : int_2
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