Nuprl Lemma : qadd-qmin
∀[a,b,c:ℚ]. ((a + qmin(b;c)) = qmin(a + b;a + c) ∈ ℚ)
Proof
Definitions occuring in Statement :
qmin: qmin(x;y),
qadd: r + s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qmin: qmin(x;y),
true: True,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
squash: ↓T,
prop: ℙ,
not: ¬A,
false: False,
guard: {T},
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rationals_wf,
q_le_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
qle_wf,
qadd_wf,
bnot_wf,
not_wf,
uiff_transitivity2,
eqtt_to_assert,
assert-q_le-eq,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
equal_wf,
qadd_preserves_qle
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,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
dependent_functionElimination,
voidElimination
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. ((a + qmin(b;c)) = qmin(a + b;a + c))
Date html generated:
2018_05_21-PM-11_56_37
Last ObjectModification:
2017_07_26-PM-06_47_05
Theory : rationals
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