Nuprl Lemma : qmin-symmetry
∀[x,y:ℚ]. (qmin(x;y) = qmin(y;x) ∈ ℚ)
Proof
Definitions occuring in Statement :
qmin: qmin(x;y),
rationals: ℚ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
all: ∀x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
qmin: qmin(x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
qless_irreflexivity,
qless_transitivity,
qle_complement_qorder,
iff_weakening_equal,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert-q_le-eq,
eqtt_to_assert,
uiff_transitivity2,
istype-void,
istype-assert,
not_wf,
bnot_wf,
qle_antisymmetry,
qle_wf,
assert_wf,
bool_wf,
equal-wf-T-base,
q_le_wf
Rules used in proof :
dependent_functionElimination,
equalityIstype,
voidElimination,
independent_pairFormation,
productElimination,
independent_functionElimination,
equalityElimination,
unionElimination,
lambdaFormation_alt,
functionIsType,
universeIsType,
independent_isectElimination,
baseClosed,
equalitySymmetry,
equalityTransitivity,
extract_by_obid,
inhabitedIsType,
isectIsTypeImplies,
axiomEquality,
hypothesisEquality,
thin,
isectElimination,
isect_memberEquality_alt,
sqequalHypSubstitution,
sqequalRule,
because_Cache,
hypothesis,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[x,y:\mBbbQ{}]. (qmin(x;y) = qmin(y;x))
Date html generated:
2019_10_29-AM-07_43_36
Last ObjectModification:
2019_10_18-PM-00_56_06
Theory : rationals
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