Nuprl Lemma : rmul-nonzero
∀x,y:ℝ. (x * y ≠ r0 ⇐⇒ x ≠ r0 ∧ y ≠ r0)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
guard: {T},
rev_implies: P ⇐ Q,
uimplies: b supposing a,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul-one-both,
rdiv-zero,
rmul-rdiv-cancel,
rmul-ac,
req_transitivity,
rmul_comm,
rmul_functionality,
rmul-assoc,
req_inversion,
req_functionality,
uiff_transitivity,
rmul-int-rdiv,
rmul-rdiv-cancel2,
rmul-zero-both,
rless_functionality,
req_weakening,
req_wf,
rless-int,
rmul_reverses_rless,
rdiv_wf,
rmul_reverses_rless_iff,
real_wf,
and_wf,
rmul-neq-zero,
rmul_wf,
rneq_wf,
rmul-is-positive,
int-to-real_wf,
rless_wf,
rmul-is-negative
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
lemma_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
inlFormation,
isectElimination,
natural_numberEquality,
productElimination,
sqequalRule,
inrFormation,
because_Cache,
independent_isectElimination,
introduction,
imageMemberEquality,
baseClosed,
multiplyEquality,
addLevel,
promote_hyp
Latex:
\mforall{}x,y:\mBbbR{}. (x * y \mneq{} r0 \mLeftarrow{}{}\mRightarrow{} x \mneq{} r0 \mwedge{} y \mneq{} r0)
Date html generated:
2016_05_18-AM-07_33_46
Last ObjectModification:
2016_01_17-AM-02_02_16
Theory : reals
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