Nuprl Lemma : ratio-functional-equation
∀[T:Type]
∀t:T. ∀F:T ⟶ T ⟶ ℝ.
(∀x,y,z:T. (((F x y) * (F y z)) = (F x z))
⇐⇒ (∀x,y:T. ((F x y) = r0)) ∨ (∃f:T ⟶ {x:ℝ| x ≠ r0} . ∀x,y:T. ((F x y) = (f x/f y))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
sq_stable: SqStable(P),
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
exists: ∃x:A. B[x],
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rneq: x ≠ y,
uiff: uiff(P;Q),
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
or: P ∨ Q,
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
rmul-rinv3,
req_inversion,
rmul-zero-both,
rmul-zero,
equal_wf,
sq_stable__rneq,
set_wf,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv,
rmul_functionality,
req_transitivity,
rmul-one,
req-iff-rsub-is-0,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rmul_preserves_req,
rless_functionality,
rless-int,
rless_wf,
rmul-is-positive,
req_weakening,
req_functionality,
square-req-self-iff,
rdiv_wf,
rneq_wf,
real_wf,
exists_wf,
int-to-real_wf,
or_wf,
req_witness,
rmul_wf,
req_wf,
all_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
imageElimination,
rename,
setElimination,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
dependent_set_memberEquality,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
inlFormation,
inrFormation,
productElimination,
dependent_functionElimination,
universeEquality,
independent_isectElimination,
setEquality,
functionEquality,
natural_numberEquality,
functionExtensionality,
because_Cache,
cumulativity,
unionElimination,
independent_functionElimination,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
\mforall{}t:T. \mforall{}F:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbR{}.
(\mforall{}x,y,z:T. (((F x y) * (F y z)) = (F x z))
\mLeftarrow{}{}\mRightarrow{} (\mforall{}x,y:T. ((F x y) = r0)) \mvee{} (\mexists{}f:T {}\mrightarrow{} \{x:\mBbbR{}| x \mneq{} r0\} . \mforall{}x,y:T. ((F x y) = (f x/f y))))
Date html generated:
2018_05_22-PM-03_12_21
Last ObjectModification:
2018_05_20-PM-11_56_45
Theory : reals_2
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