Nuprl Lemma : qabs-qdiv
∀[r:ℚ]. ∀[s:{s:ℚ| ¬(s = 0 ∈ ℚ)} ]. (|(r/s)| = (|r|/|s|) ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qdiv: (r/s),
rationals: ℚ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qdiv: (r/s),
not: ¬A,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
assert-qeq,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-T-base,
set_wf,
rationals_wf,
qabs_wf,
qmul_wf,
qinv_wf,
equal_wf,
squash_wf,
true_wf,
qabs-qinv,
iff_weakening_equal,
qabs-qmul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
because_Cache,
addLevel,
sqequalHypSubstitution,
impliesFunctionality,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
applyEquality,
sqequalRule,
productElimination,
independent_isectElimination,
hypothesisEquality,
lambdaEquality,
baseClosed,
isect_memberEquality,
axiomEquality,
dependent_set_memberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
independent_functionElimination
Latex:
\mforall{}[r:\mBbbQ{}]. \mforall{}[s:\{s:\mBbbQ{}| \mneg{}(s = 0)\} ]. (|(r/s)| = (|r|/|s|))
Date html generated:
2018_05_21-PM-11_57_01
Last ObjectModification:
2017_07_26-PM-06_47_26
Theory : rationals
Home
Index