Nuprl Lemma : rabs-rdiv
∀x,y:ℝ. (y ≠ r0 ⇒ (|(x/y)| = (|x|/|y|)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
rdiv: (x/y),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rabs-neq-zero,
rneq_wf,
int-to-real_wf,
real_wf,
rabs_wf,
rmul_wf,
rinv_wf2,
rless_wf,
req_weakening,
req_functionality,
req_transitivity,
rabs-rmul,
rmul_functionality,
rabs-rinv
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
natural_numberEquality,
inrFormation,
because_Cache,
independent_isectElimination,
productElimination
Latex:
\mforall{}x,y:\mBbbR{}. (y \mneq{} r0 {}\mRightarrow{} (|(x/y)| = (|x|/|y|)))
Date html generated:
2016_05_18-AM-07_26_56
Last ObjectModification:
2015_12_28-AM-00_50_22
Theory : reals
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