Nuprl Lemma : qavg-eq-iff-3
∀[a,b:ℚ].  uiff(qavg(b;a) = 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}
, 
prop: ℙ
, 
squash: ↓T
, 
true: True
, 
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, 
qmul-preserves-eq, 
qmul_one_qrng, 
q_distrib, 
qadd_inv_assoc_q, 
qadd_ac_1_q, 
iff_weakening_equal, 
qmul-qdiv-cancel, 
subtype_rel_self, 
qadd_comm_q, 
equal-wf-T-base, 
not_wf, 
istype-universe, 
true_wf, 
squash_wf, 
equal_wf, 
int-subtype-rationals, 
qmul_wf, 
rationals_wf, 
assert-qeq, 
qadd_wf, 
qdiv_wf
Rules used in proof : 
independent_functionElimination, 
imageMemberEquality, 
universeEquality, 
instantiate, 
imageElimination, 
lambdaEquality_alt, 
minusEquality, 
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(b;a)  =  a;a  =  b)
Date html generated:
2019_10_29-AM-07_44_42
Last ObjectModification:
2019_10_21-PM-08_26_25
Theory : rationals
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