Nuprl Lemma : rless-implies
∀x,y:ℝ. ((x < y) ⇒ (∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) ⇒ (5 ≤ ((y m) - x m)))))
Proof
Definitions occuring in Statement :
rless: x < y,
real: ℝ,
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
real: ℝ,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
uiff: uiff(P;Q)
Lemmas referenced :
false_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract-is-int-iff,
multiply-is-int-iff,
mul_cancel_in_le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
real_wf,
rless_wf,
subtract_wf,
le_wf,
nat_plus_wf,
all_wf,
less_than_wf,
mul_nat_plus,
rless-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
dependent_pairFormation,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
lambdaEquality,
functionEquality,
setElimination,
rename,
applyEquality,
multiplyEquality,
unionElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} (\mexists{}n:\mBbbN{}\msupplus{}. \mforall{}m:\mBbbN{}\msupplus{}. ((n \mleq{} m) {}\mRightarrow{} (5 \mleq{} ((y m) - x m)))))
Date html generated:
2016_05_18-AM-07_15_32
Last ObjectModification:
2016_01_17-AM-01_53_55
Theory : reals
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