Nuprl Lemma : implies-equal-real
∀[x,y:ℝ].  x = y ∈ ℝ supposing ∀n:ℕ+. ((x n) = (y n) ∈ ℤ)
Proof
Definitions occuring in Statement : 
real: ℝ
, 
nat_plus: ℕ+
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
apply: f a
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
real: ℝ
, 
squash: ↓T
, 
prop: ℙ
, 
all: ∀x:A. B[x]
, 
true: True
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
sq_stable: SqStable(P)
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
equal_wf, 
squash_wf, 
true_wf, 
iff_weakening_equal, 
nat_plus_wf, 
sq_stable__regular-int-seq, 
less_than_wf, 
regular-int-seq_wf, 
all_wf, 
real_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
dependent_set_memberEquality, 
functionExtensionality, 
applyEquality, 
thin, 
lambdaEquality, 
sqequalHypSubstitution, 
imageElimination, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
equalityTransitivity, 
hypothesis, 
equalitySymmetry, 
universeEquality, 
intEquality, 
dependent_functionElimination, 
setElimination, 
rename, 
natural_numberEquality, 
sqequalRule, 
imageMemberEquality, 
baseClosed, 
independent_isectElimination, 
productElimination, 
independent_functionElimination, 
because_Cache, 
independent_pairFormation, 
isect_memberEquality, 
axiomEquality
Latex:
\mforall{}[x,y:\mBbbR{}].    x  =  y  supposing  \mforall{}n:\mBbbN{}\msupplus{}.  ((x  n)  =  (y  n))
Date html generated:
2017_10_02-PM-07_13_14
Last ObjectModification:
2017_07_28-AM-07_20_01
Theory : reals
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