Nuprl Lemma : qmin_lb
∀a,b,c:ℚ. (qmin(b;c) ≤ a ⇐⇒ (b ≤ a) ∨ (c ≤ a))
Proof
Definitions occuring in Statement :
qmin: qmin(x;y),
qle: r ≤ s,
rationals: ℚ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
qmin: qmin(x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
true: True,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
squash: ↓T
Lemmas referenced :
rationals_wf,
q_le_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
qle_wf,
qle_transitivity_qorder,
or_wf,
bnot_wf,
not_wf,
qle_complement_qorder,
qless_transitivity_2_qorder,
qle_weakening_lt_qorder,
uiff_transitivity2,
eqtt_to_assert,
assert-q_le-eq,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
because_Cache,
independent_pairFormation,
inlFormation,
unionElimination,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
inrFormation,
productElimination,
equalityElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
dependent_functionElimination
Latex:
\mforall{}a,b,c:\mBbbQ{}. (qmin(b;c) \mleq{} a \mLeftarrow{}{}\mRightarrow{} (b \mleq{} a) \mvee{} (c \mleq{} a))
Date html generated:
2018_05_21-PM-11_55_18
Last ObjectModification:
2017_07_26-PM-06_46_06
Theory : rationals
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