Nuprl Lemma : fractions-req
∀[a,b,c,d:ℝ]. (c ≠ r0 ⇒ d ≠ r0 ⇒ uiff((a/c) = (b/d);(a * d) = (b * c)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rmul_wf,
req_wf,
rdiv_wf,
rneq_wf,
int-to-real_wf,
real_wf,
rmul_preserves_req,
req_weakening,
req_functionality,
rmul_functionality,
rmul-rdiv-cancel2,
uiff_transitivity,
req_inversion,
rmul-assoc,
req_transitivity,
rmul-ac,
rmul_comm,
rmul-rdiv-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,b,c,d:\mBbbR{}]. (c \mneq{} r0 {}\mRightarrow{} d \mneq{} r0 {}\mRightarrow{} uiff((a/c) = (b/d);(a * d) = (b * c)))
Date html generated:
2017_10_03-AM-08_38_43
Last ObjectModification:
2017_03_27-AM-01_00_51
Theory : reals
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