Nuprl Lemma : rinv-of-rmul
∀[x,y:ℝ]. (rinv(x * y) = (rinv(x) * rinv(y))) supposing (y ≠ r0 and x ≠ r0)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rinv: rinv(x),
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul-neq-zero,
req_inversion,
rmul_wf,
rinv_wf2,
req_witness,
rneq_wf,
int-to-real_wf,
real_wf,
rmul-inverse-is-rinv,
req_functionality,
rmul-ac,
req_weakening,
req_wf,
uiff_transitivity,
rmul_functionality,
req_transitivity,
rmul_assoc,
rmul-rinv,
rmul-one,
rmul-rinv2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. (rinv(x * y) = (rinv(x) * rinv(y))) supposing (y \mneq{} r0 and x \mneq{} r0)
Date html generated:
2016_05_18-AM-07_12_12
Last ObjectModification:
2015_12_28-AM-00_40_25
Theory : reals
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