Nuprl Lemma : radd-rneq0
∀x,y:ℝ. (x + y ≠ r0 ⇐⇒ x ≠ -(y))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rminus: -(x),
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
or: P ∨ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rneq_wf,
radd_wf,
int-to-real_wf,
rminus_wf,
real_wf,
radd-preserves-rless,
rless_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
itermConstant_wf,
rless_functionality,
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_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
inhabitedIsType,
unionElimination,
inlFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
inrFormation_alt,
because_Cache,
independent_isectElimination,
sqequalRule,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}x,y:\mBbbR{}. (x + y \mneq{} r0 \mLeftarrow{}{}\mRightarrow{} x \mneq{} -(y))
Date html generated:
2019_10_29-AM-09_57_53
Last ObjectModification:
2019_04_01-PM-05_33_11
Theory : reals
Home
Index