Nuprl Lemma : qmul-zero-div
∀[a:ℚ]. ∀[b:ℤ-o]. (((0/b) * a) = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
int_nzero: ℤ-o,
qdiv: (r/s),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
squash: ↓T,
and: P ∧ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
uiff: uiff(P;Q)
Lemmas referenced :
equal-wf-T-base,
rationals_wf,
qmul_wf,
int_nzero_wf,
int_nzero_properties,
qinv_wf,
int-subtype-rationals,
equal_wf,
squash_wf,
true_wf,
qmul_zero_qrng,
iff_weakening_equal,
int-equal-in-rationals,
equal-wf-base,
int_subtype_base,
not_wf,
assert-qeq,
assert_wf,
qeq_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
hyp_replacement,
hypothesis,
equalitySymmetry,
applyLambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
baseClosed,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
setElimination,
rename,
applyEquality,
independent_isectElimination,
natural_numberEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
productElimination,
imageMemberEquality,
independent_functionElimination,
intEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}[a:\mBbbQ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. (((0/b) * a) = 0)
Date html generated:
2018_05_21-PM-11_50_44
Last ObjectModification:
2017_07_26-PM-06_44_06
Theory : rationals
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