Nuprl Lemma : qavg-eq-iff-2
∀[a,b:ℚ]. uiff(qavg(a;b) = a ∈ ℚ;a = b ∈ ℚ)
Proof
Definitions occuring in Statement :
qavg: qavg(a;b),
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
qadd: r + s,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
prop: ℙ,
squash: ↓T,
false: False,
assert: ↑b,
bfalse: ff,
eq_int: (i =z j),
btrue: tt,
ifthenelse: if b then t else f fi ,
evalall: evalall(t),
callbyvalueall: callbyvalueall,
qeq: qeq(r;s),
implies: P ⇒ Q,
not: ¬A,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
qavg: qavg(a;b)
Lemmas referenced :
qmul_ident,
equal-wf-T-base,
not_wf,
qmul-preserves-eq,
qmul_one_qrng,
q_distrib,
qadd_inv_assoc_q,
qadd_ac_1_q,
iff_weakening_equal,
subtype_rel_self,
int-subtype-rationals,
qmul-qdiv-cancel,
istype-universe,
true_wf,
squash_wf,
equal_wf,
qmul_wf,
rationals_wf,
assert-qeq,
qadd_wf,
qdiv_wf
Rules used in proof :
minusEquality,
independent_functionElimination,
imageMemberEquality,
universeEquality,
instantiate,
imageElimination,
lambdaEquality_alt,
applyLambdaEquality,
universeIsType,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
independent_pairEquality,
sqequalBase,
baseClosed,
voidElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
lambdaFormation_alt,
independent_isectElimination,
because_Cache,
applyEquality,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesisEquality,
inhabitedIsType,
equalityIstype,
hypothesis,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(qavg(a;b) = a;a = b)
Date html generated:
2019_10_29-AM-07_44_36
Last ObjectModification:
2019_10_21-PM-06_52_21
Theory : rationals
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