Nuprl Lemma : qmin-as-qmax
∀[x,y:ℚ]. (qmin(x;y) = -(qmax(-(x);-(y))) ∈ ℚ)
Proof
Definitions occuring in Statement :
qmin: qmin(x;y),
qmax: qmax(x;y),
qmul: r * s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qmin: qmin(x;y),
qmax: qmax(x;y),
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uiff: uiff(P;Q),
false: False,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ
Lemmas referenced :
rationals_wf,
q_le_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
qle_wf,
qmul_wf,
equal_wf,
qinv_inv_q,
iff_weakening_equal,
qadd_preserves_qle,
qinverse_q,
qadd_wf,
qle_antisymmetry,
bnot_wf,
not_wf,
qle_complement_qorder,
qadd_preserves_qless,
qless_wf,
qless_transitivity,
qless_irreflexivity,
uiff_transitivity2,
eqtt_to_assert,
assert-q_le-eq,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
qadd_comm_q,
qadd_ac_1_q,
mon_ident_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseClosed,
minusEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
voidElimination,
lambdaFormation,
unionElimination,
equalityElimination,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[x,y:\mBbbQ{}]. (qmin(x;y) = -(qmax(-(x);-(y))))
Date html generated:
2018_05_21-PM-11_58_28
Last ObjectModification:
2017_07_26-PM-06_48_06
Theory : rationals
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