Nuprl Lemma : qabs-zero
∀[r:ℚ]. uiff(r = 0 ∈ ℚ;|r| = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
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,
subtype_rel: A ⊆r B,
has-value: (a)↓,
has-valueall: has-valueall(a),
all: ∀x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal-wf-T-base,
rationals_wf,
qabs_wf,
int-subtype-rationals,
valueall-type-has-valueall,
rationals-valueall-type,
evalall-reduce,
qpositive_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
equal_wf,
squash_wf,
true_wf,
qinv_inv_q,
iff_weakening_equal,
qinv_id_q,
qmul_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
baseClosed,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
hyp_replacement,
applyLambdaEquality,
independent_isectElimination,
callbyvalueReduce,
dependent_functionElimination,
unionElimination,
instantiate,
cumulativity,
independent_functionElimination,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
minusEquality
Latex:
\mforall{}[r:\mBbbQ{}]. uiff(r = 0;|r| = 0)
Date html generated:
2018_05_21-PM-11_51_41
Last ObjectModification:
2017_07_26-PM-06_44_38
Theory : rationals
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