Nuprl Lemma : qmin-eq-iff-1
∀[q,r:ℚ]. uiff(qmin(q;r) = q ∈ ℚ;q ≤ r)
Proof
Definitions occuring in Statement :
qmin: qmin(x;y),
qle: r ≤ s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
prop: ℙ,
implies: P ⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
qle_weakening_eq_qorder,
qle_weakening_lt_qorder,
not-qle,
decidable__qle,
qmin-eq-iff,
iff_weakening_uiff,
rev_implies_wf,
member_wf,
qmin_wf,
equal_wf,
qle_antisymmetry,
qle_wf,
rationals_wf,
qle_witness
Rules used in proof :
unionElimination,
promote_hyp,
functionEquality,
productEquality,
functionIsTypeImplies,
dependent_functionElimination,
lambdaEquality_alt,
independent_isectElimination,
lambdaFormation_alt,
equalityIstype,
functionIsType,
productIsType,
independent_pairFormation,
universeIsType,
because_Cache,
axiomEquality,
inhabitedIsType,
isectIsTypeImplies,
hypothesis,
independent_functionElimination,
extract_by_obid,
hypothesisEquality,
isectElimination,
isect_memberEquality_alt,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[q,r:\mBbbQ{}]. uiff(qmin(q;r) = q;q \mleq{} r)
Date html generated:
2019_10_29-AM-07_44_13
Last ObjectModification:
2019_10_21-PM-06_23_33
Theory : rationals
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