Nuprl Lemma : qabs-qminus
∀[r:ℚ]. (|r| = |-(r)| ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qmul: r * s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
true: True,
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
qabs: |r|,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
qpositive: qpositive(r),
btrue: tt,
lt_int: i <z j,
bfalse: ff,
qmul: r * s
Lemmas referenced :
uall_wf,
squash_wf,
true_wf,
rationals_wf,
equal_wf,
qabs_wf,
qabs-qmul,
int-subtype-rationals,
iff_weakening_equal,
qmul_one_qrng
Rules used in proof :
cut,
applyEquality,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
sqequalRule,
because_Cache,
minusEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberFormation
Latex:
\mforall{}[r:\mBbbQ{}]. (|r| = |-(r)|)
Date html generated:
2018_05_21-PM-11_53_10
Last ObjectModification:
2017_07_26-PM-06_45_30
Theory : rationals
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