Nuprl Lemma : rfun*2_functionality
∀[f:ℝ ⟶ ℝ ⟶ ℝ]. ∀[x,y,u,v:ℝ*].
((∀[a,b,c,d:ℝ]. (f a c) = (f b d) supposing (a = b) ∧ (c = d)) ⇒ x = y ⇒ u = v ⇒ f*(x;u) = f*(y;v))
Proof
Definitions occuring in Statement :
rfun*2: f*(x;y),
req*: x = y,
real*: ℝ*,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
req*: x = y,
exists: ∃x:A. B[x],
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
rfun*2: f*(x;y),
real*: ℝ*,
subtype_rel: A ⊆r B,
int_upper: {i...},
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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_nat,
int_upper_properties,
req_witness,
rfun*2_wf,
int_upper_wf,
all_wf,
req_wf,
req*_wf,
uall_wf,
real_wf,
isect_wf,
real*_wf,
int_upper_subtype_int_upper,
imax_ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
because_Cache,
functionExtensionality,
productEquality,
functionEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[f:\mBbbR{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[x,y,u,v:\mBbbR{}*].
((\mforall{}[a,b,c,d:\mBbbR{}]. (f a c) = (f b d) supposing (a = b) \mwedge{} (c = d))
{}\mRightarrow{} x = y
{}\mRightarrow{} u = v
{}\mRightarrow{} f*(x;u) = f*(y;v))
Date html generated:
2018_05_22-PM-03_15_48
Last ObjectModification:
2017_10_06-PM-02_33_25
Theory : reals_2
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