Nuprl Lemma : rmetric-meq
∀[x,y:ℝ]. uiff(x ≡ y;x = y)
Proof
Definitions occuring in Statement :
rmetric: rmetric(),
meq: x ≡ y,
req: x = y,
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rmetric: rmetric(),
meq: x ≡ y,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
rev_uimplies: rev_uimplies(P;Q),
absval: |i|,
req_int_terms: t1 ≡ t2
Lemmas referenced :
rabs-difference-is-zero,
req_witness,
req_wf,
rabs_wf,
rsub_wf,
int-to-real_wf,
real_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-int,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
req_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
req_transitivity,
rabs-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
isectElimination,
universeIsType,
natural_numberEquality,
independent_pairEquality,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
minusEquality,
independent_isectElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
voidElimination,
int_eqEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. uiff(x \mequiv{} y;x = y)
Date html generated:
2019_10_29-AM-11_03_33
Last ObjectModification:
2019_10_02-AM-09_44_19
Theory : reals
Home
Index