Nuprl Lemma : radd-preserves-rneq
∀x,y,z:ℝ. (x ≠ y ⇐⇒ z + x ≠ z + y)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
radd: a + b,
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
radd-preserves-rless,
rless_wf,
radd_wf,
rneq_wf,
rless-implies-rless,
real_wf,
rsub_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
inlFormation_alt,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
inrFormation_alt,
independent_isectElimination,
inhabitedIsType,
natural_numberEquality,
because_Cache,
sqequalRule,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}x,y,z:\mBbbR{}. (x \mneq{} y \mLeftarrow{}{}\mRightarrow{} z + x \mneq{} z + y)
Date html generated:
2019_10_29-AM-09_35_45
Last ObjectModification:
2019_04_01-PM-05_44_31
Theory : reals
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