Nuprl Lemma : radd-of-nonneg-is-zero
∀[a,b:{x:ℝ| r0 ≤ x} ]. uiff((a + b) = r0;(a = r0) ∧ (b = r0))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rleq_antisymmetry,
int-to-real_wf,
sq_stable__rleq,
req_witness,
req_wf,
radd_wf,
real_wf,
rleq_wf,
radd-zero,
req_functionality,
radd_functionality,
req_weakening,
radd-preserves-req,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
radd-preserves-rleq,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
independent_isectElimination,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_pairEquality,
universeIsType,
productIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
setIsType,
dependent_functionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination
Latex:
\mforall{}[a,b:\{x:\mBbbR{}| r0 \mleq{} x\} ]. uiff((a + b) = r0;(a = r0) \mwedge{} (b = r0))
Date html generated:
2019_10_29-AM-09_35_36
Last ObjectModification:
2019_05_14-PM-03_33_41
Theory : reals
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