Nuprl Lemma : rrel*_functionality
∀[R:ℝ ⟶ ℝ ⟶ ℙ]
((∀x1,x2,y1,y2:ℝ. ((x1 = x2) ⇒ (y1 = y2) ⇒ (R x1 y1 ⇐⇒ R x2 y2)))
⇒ (∀x1,x2,y1,y2:ℝ*. (x1 = x2 ⇒ y1 = y2 ⇒ (R*(x1,y1) ⇐⇒ R*(x2,y2)))))
Proof
Definitions occuring in Statement :
req*: x = y,
rrel*: R*(x,y),
real*: ℝ*,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
req*: x = y,
exists: ∃x:A. B[x],
rrel*: R*(x,y),
member: t ∈ T,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
int_upper: {i...},
real*: ℝ*,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
imax_wf,
imax_nat,
nat_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
int_upper_subtype_int_upper,
imax_ub,
int_upper_properties,
int_upper_subtype_nat,
int_upper_wf,
all_wf,
real_wf,
rrel*_wf,
req*_wf,
real*_wf,
req_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
because_Cache,
inrFormation,
inlFormation,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[R:\mBbbR{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbP{}]
((\mforall{}x1,x2,y1,y2:\mBbbR{}. ((x1 = x2) {}\mRightarrow{} (y1 = y2) {}\mRightarrow{} (R x1 y1 \mLeftarrow{}{}\mRightarrow{} R x2 y2)))
{}\mRightarrow{} (\mforall{}x1,x2,y1,y2:\mBbbR{}*. (x1 = x2 {}\mRightarrow{} y1 = y2 {}\mRightarrow{} (R*(x1,y1) \mLeftarrow{}{}\mRightarrow{} R*(x2,y2)))))
Date html generated:
2018_05_22-PM-03_14_51
Last ObjectModification:
2017_10_06-PM-06_05_14
Theory : reals_2
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