Nuprl Lemma : rmul-neq-zero
∀x,y:ℝ. (x ≠ r0 ⇒ y ≠ r0 ⇒ x * 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],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
rneq: x ≠ y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
real: ℝ,
guard: {T},
uimplies: b supposing a,
rsub: x - y
Lemmas referenced :
rless-iff-rpositive,
int-to-real_wf,
rmul_wf,
rless_wf,
or_wf,
rpositive_wf,
rsub_wf,
real_wf,
radd_wf,
rminus_wf,
rpositive-rmul,
rpositive_functionality,
rmul_over_rminus,
radd-zero-both,
radd_comm,
radd_functionality,
rminus-zero,
req_weakening,
req_transitivity,
rminus_functionality,
rminus-rminus
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
natural_numberEquality,
hypothesis,
productElimination,
independent_functionElimination,
addLevel,
orFunctionality,
applyEquality,
lambdaEquality,
setElimination,
rename,
because_Cache,
inrFormation,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inlFormation,
promote_hyp,
levelHypothesis,
orLevelFunctionality
Latex:
\mforall{}x,y:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} y \mneq{} r0 {}\mRightarrow{} x * y \mneq{} r0)
Date html generated:
2017_10_03-AM-08_27_46
Last ObjectModification:
2017_03_01-AM-00_03_32
Theory : reals
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