Nuprl Lemma : qabs-qmul
∀[r,s:ℚ]. (|r * s| = (|r| * |s|) ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qmul: 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,
all: ∀x:A. B[x],
qabs: |r|,
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
guard: {T},
rev_implies: P ⇐ Q,
bfalse: ff,
not: ¬A,
true: True,
false: False
Lemmas referenced :
q_trichotomy,
valueall-type-has-valueall,
rationals-valueall-type,
evalall-reduce,
qmul_wf,
qpositive_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
qless_wf,
int-subtype-rationals,
eqtt_to_assert,
assert-qpositive,
equal_wf,
ite_rw_true,
qmul-positive,
iff_weakening_equal,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
qmul_over_minus_qrng,
or_wf,
qminus-positive,
qless_transitivity,
qless_irreflexivity,
qmul_ac_1_qrng,
qmul_assoc_qrng,
qmul_assoc,
qinv_inv_q,
rationals_wf,
squash_wf,
true_wf,
qmul_zero_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
callbyvalueReduce,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
natural_numberEquality,
applyEquality,
independent_functionElimination,
productElimination,
lambdaEquality,
imageElimination,
inlFormation,
independent_pairFormation,
productEquality,
minusEquality,
imageMemberEquality,
impliesFunctionality,
addLevel,
orFunctionality,
promote_hyp,
andLevelFunctionality,
voidElimination,
isect_memberEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality,
universeEquality,
inrFormation
Latex:
\mforall{}[r,s:\mBbbQ{}]. (|r * s| = (|r| * |s|))
Date html generated:
2018_05_21-PM-11_53_06
Last ObjectModification:
2017_07_26-PM-06_45_27
Theory : rationals
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