Nuprl Lemma : req-rdiv
∀x,y,z:ℝ. (z ≠ r0 ⇒ (x = (y/z) ⇐⇒ (x * z) = y))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
req: 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 :
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
false: False,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
true: True,
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
pi2: snd(t),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
real_wf,
int-to-real_wf,
rneq_wf,
rmul_wf,
rdiv_wf,
req_wf,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
istype-int,
req_functionality,
rmul_functionality,
req_weakening,
rdiv_functionality,
req_inversion
Rules used in proof :
natural_numberEquality,
hypothesis,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
because_Cache,
lambdaEquality_alt,
int_eqEquality,
approximateComputation,
sqequalRule,
productElimination
Latex:
\mforall{}x,y,z:\mBbbR{}. (z \mneq{} r0 {}\mRightarrow{} (x = (y/z) \mLeftarrow{}{}\mRightarrow{} (x * z) = y))
Date html generated:
2019_10_29-AM-09_56_16
Last ObjectModification:
2019_04_01-PM-11_09_44
Theory : reals
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