Nuprl Lemma : qmul_preserves_qle
∀[a,b,c:ℚ]. uiff(a ≤ b;(c * a) ≤ (c * b)) supposing 0 < c
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qmul: r * s,
rationals: ℚ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
not: ¬A,
guard: {T},
false: False
Lemmas referenced :
qless_wf,
qmul_wf,
iff_weakening_uiff,
qle_wf,
equal_wf,
rationals_wf,
qle-iff,
qle_witness,
qmul_preserves_qless,
int-subtype-rationals,
sq_stable_from_decidable,
decidable__or,
decidable__qless,
decidable__equal_rationals,
qmul-preserves-eq,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation_alt,
sqequalRule,
unionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
equalityIstype,
inhabitedIsType,
because_Cache,
productElimination,
independent_isectElimination,
unionEquality,
independent_functionElimination,
dependent_functionElimination,
promote_hyp,
unionElimination,
inlFormation_alt,
inrFormation_alt,
applyEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
closedConclusion,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
lambdaFormation_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. uiff(a \mleq{} b;(c * a) \mleq{} (c * b)) supposing 0 < c
Date html generated:
2020_05_20-AM-09_16_17
Last ObjectModification:
2020_02_26-AM-09_59_19
Theory : rationals
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