Nuprl Lemma : clear-denominator2
∀[a,b,c,d,e:ℝ]. uiff((((a/b) * c) * e) = d;(c * e * a) = (d * b)) supposing b ≠ r0
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),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rmul_wf,
req_wf,
rdiv_wf,
uiff_wf,
rneq_wf,
int-to-real_wf,
real_wf,
iff_weakening_uiff,
clear-denominator1,
req_functionality,
rmul_assoc,
req_weakening,
uiff_transitivity,
req_inversion,
rmul-assoc,
rmul_functionality,
rmul_comm,
rmul-ac
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
independent_isectElimination,
cumulativity,
natural_numberEquality,
addLevel,
productElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,b,c,d,e:\mBbbR{}]. uiff((((a/b) * c) * e) = d;(c * e * a) = (d * b)) supposing b \mneq{} r0
Date html generated:
2017_10_03-AM-08_38_34
Last ObjectModification:
2017_03_14-AM-11_43_19
Theory : reals
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