Nuprl Lemma : rmul_preserves_rneq
∀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],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
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,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q)
Lemmas referenced :
rneq_wf,
int-to-real_wf,
real_wf,
rmul_preserves_rless,
rless_wf,
rmul_wf,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rsub_wf,
rless_functionality,
req_transitivity,
itermConstant_wf,
rmul_functionality,
rmul-identity1,
req_weakening,
rmul_reverses_rless,
rmul_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
lemma_by_obid,
inrFormation,
sqequalRule,
productElimination,
independent_isectElimination,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
inlFormation,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}a,b,x:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} x * a \mneq{} x * b)
Date html generated:
2017_10_03-AM-08_28_32
Last ObjectModification:
2017_07_28-AM-07_25_18
Theory : reals
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