Nuprl Lemma : rneq-rmul
∀x,y,a,b:ℝ. (x * y ≠ a * b ⇒ (x ≠ a ∨ y ≠ b))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rmul: a * b,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
or: P ∨ Q,
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rless_wf,
int-to-real_wf,
rabs_wf,
rsub_wf,
rmul_wf,
real_wf,
rneq-iff-rabs,
rneq_wf,
all_wf,
or_wf,
rleq_functionality_wrt_implies,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
rless_functionality_wrt_implies,
radd-positive-implies,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rless_functionality,
req_weakening,
rabs_functionality,
rabs-rmul,
rmul-is-positive,
zero-rleq-rabs,
rless_transitivity1,
rless_irreflexivity
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
addLevel,
allFunctionality,
impliesFunctionality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
orFunctionality,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
inlFormation,
inrFormation,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
promote_hyp
Latex:
\mforall{}x,y,a,b:\mBbbR{}. (x * y \mneq{} a * b {}\mRightarrow{} (x \mneq{} a \mvee{} y \mneq{} b))
Date html generated:
2017_10_03-AM-08_47_26
Last ObjectModification:
2017_06_21-PM-03_25_55
Theory : reals
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