Nuprl Lemma : rinv_preserves_rneq
∀a,b:ℝ. (a ≠ r0 ⇒ b ≠ r0 ⇒ a ≠ b ⇒ (r1/a) ≠ (r1/b))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
guard: {T},
rdiv: (x/y),
cand: A c∧ B
Lemmas referenced :
rneq_wf,
int-to-real_wf,
real_wf,
rmul_reverses_rless,
rless-int,
rless_wf,
rmul_wf,
rless_functionality,
rminus_wf,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rinv_preserves_rless,
rless-implies-rless,
rsub_wf,
rdiv_wf,
rmul_reverses_rless_iff,
rinv_wf2,
req_transitivity,
rminus_functionality,
rinv-as-rdiv,
rneq-symmetry,
rminus-neq-zero,
rminus-rdiv2,
rmul-rinv,
rmul_preserves_rless,
rless_transitivity2,
rleq_weakening_rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
minusEquality,
independent_functionElimination,
productElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_isectElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
inrFormation,
inlFormation
Latex:
\mforall{}a,b:\mBbbR{}. (a \mneq{} r0 {}\mRightarrow{} b \mneq{} r0 {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} (r1/a) \mneq{} (r1/b))
Date html generated:
2017_10_03-AM-08_36_54
Last ObjectModification:
2017_07_28-AM-07_29_44
Theory : reals
Home
Index