Nuprl Lemma : qabs-squared
∀[r:ℚ]. ((|r| * |r|) = (r * r) ∈ 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],
qabs: |r|,
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a)
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
qpositive_wf,
bool_wf,
eqtt_to_assert,
qmul_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
qmul_assoc,
int-subtype-rationals,
iff_weakening_equal,
qmul_ac_1_qrng,
qinv_inv_q,
evalall-reduce
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
hypothesis,
because_Cache,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
independent_isectElimination,
hypothesisEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
sqequalRule,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
minusEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
callbyvalueReduce
Latex:
\mforall{}[r:\mBbbQ{}]. ((|r| * |r|) = (r * r))
Date html generated:
2018_05_21-PM-11_51_47
Last ObjectModification:
2017_07_26-PM-06_44_41
Theory : rationals
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