Nuprl Lemma : qmul-zero
∀a,b:ℚ. ((a * b) = 0 ∈ ℚ ⇐⇒ (a = 0 ∈ ℚ) ∨ (b = 0 ∈ ℚ))
Proof
Definitions occuring in Statement :
qmul: r * s,
rationals: ℚ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
uimplies: b supposing a,
not: ¬A,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
eq_int: (i =z j),
bfalse: ff,
assert: ↑b,
false: False
Lemmas referenced :
equal-wf-T-base,
rationals_wf,
qmul_wf,
or_wf,
decidable__equal_rationals,
qinv_wf,
assert-qeq,
assert_wf,
qeq_wf2,
not_wf,
int-subtype-rationals,
equal_wf,
qmul_ident,
iff_weakening_equal,
squash_wf,
true_wf,
qmul_inv,
qmul_zero_qrng,
qmul_assoc_qrng,
qmul_ac_1_qrng,
qmul_comm_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
baseClosed,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
inlFormation,
sqequalRule,
inrFormation,
hyp_replacement,
applyLambdaEquality,
independent_isectElimination,
addLevel,
impliesFunctionality,
productElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
independent_functionElimination,
universeEquality,
voidElimination
Latex:
\mforall{}a,b:\mBbbQ{}. ((a * b) = 0 \mLeftarrow{}{}\mRightarrow{} (a = 0) \mvee{} (b = 0))
Date html generated:
2018_05_21-PM-11_56_03
Last ObjectModification:
2017_07_26-PM-06_46_36
Theory : rationals
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