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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