Nuprl Lemma : mul_preserves_le2
∀[a,c:ℕ]. ∀[b,d:ℤ]. ((a * c) ≤ (b * d)) supposing ((a ≤ b) and (c ≤ d))
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,
nat: ℕ,
and: P ∧ Q,
le: A ≤ B,
cand: A c∧ B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
top: Top
Lemmas referenced :
mul_preserves_le,
sq_stable_from_decidable,
le_wf,
decidable__le,
le_transitivity,
istype-le,
le_witness_for_triv,
istype-int,
istype-nat,
istype-void,
mul-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
Error :dependent_set_memberEquality_alt,
setElimination,
rename,
independent_pairFormation,
productElimination,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
voidElimination
Latex:
\mforall{}[a,c:\mBbbN{}]. \mforall{}[b,d:\mBbbZ{}]. ((a * c) \mleq{} (b * d)) supposing ((a \mleq{} b) and (c \mleq{} d))
Date html generated:
2019_06_20-AM-11_23_20
Last ObjectModification:
2019_02_08-PM-01_26_03
Theory : arithmetic
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