Nuprl Lemma : rmul_preserves_rneq_iff
∀a,b,x:ℝ. (x ≠ r0 ⇒ (a ≠ b ⇐⇒ x * a ≠ x * b))
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,
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
rneq: x ≠ y,
or: P ∨ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rneq_wf,
rmul_wf,
int-to-real_wf,
real_wf,
rmul_preserves_rneq,
rmul_reverses_rless_iff,
rless-implies-rless,
rless_wf,
rmul_preserves_rless,
rsub_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
inrFormation,
productElimination,
independent_isectElimination,
inlFormation,
because_Cache,
sqequalRule,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}a,b,x:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} (a \mneq{} b \mLeftarrow{}{}\mRightarrow{} x * a \mneq{} x * b))
Date html generated:
2017_10_03-AM-08_35_16
Last ObjectModification:
2017_06_16-PM-00_46_33
Theory : reals
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