Nuprl Lemma : qmul-not-zero
∀[a,b:ℚ]. uiff(¬((a * b) = 0 ∈ ℚ);(¬(a = 0 ∈ ℚ)) ∧ (¬(b = 0 ∈ ℚ)))
Proof
Definitions occuring in Statement :
qmul: r * s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
qmul_zero_qrng,
equal-wf-T-base,
qmul_wf,
rationals_wf,
not_wf,
qdiv_wf,
int-subtype-rationals,
equal_wf,
squash_wf,
true_wf,
qmul-qdiv-cancel4,
qmul_one_qrng,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
sqequalHypSubstitution,
independent_functionElimination,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
productElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
because_Cache,
voidElimination,
baseClosed,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
productEquality,
isect_memberEquality,
equalityTransitivity,
natural_numberEquality,
applyEquality,
independent_isectElimination,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(\mneg{}((a * b) = 0);(\mneg{}(a = 0)) \mwedge{} (\mneg{}(b = 0)))
Date html generated:
2018_05_21-PM-11_51_26
Last ObjectModification:
2017_07_26-PM-06_44_31
Theory : rationals
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