Nuprl Lemma : rnonzero-lemma1
∀[x:ℝ]. ∀[n:ℕ+]. ∀m:ℕ+. ((n ≤ m) ⇒ (m ≤ (n * |x m|))) supposing 5 ≤ |x n|
Proof
Definitions occuring in Statement :
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
real: ℝ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
nat: ℕ,
sq_stable: SqStable(P),
regular-int-seq: k-regular-seq(f),
squash: ↓T,
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
top: Top,
ge: i ≥ j ,
guard: {T},
subtract: n - m,
true: True,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable__le,
absval_wf,
nat_plus_wf,
nat_wf,
int-triangle-inequality,
subtract_wf,
le_wf,
less_than'_wf,
real_wf,
le_functionality,
le_weakening,
add_functionality_wrt_le,
mul-distributes,
mul-associates,
minus-one-mul,
mul-swap,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
add-zero,
false_wf,
squash_wf,
true_wf,
absval_mul,
add_functionality_wrt_eq,
iff_weakening_equal,
multiply_functionality_wrt_le,
equal_wf,
nat_plus_subtype_nat,
nat_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
absval_pos
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
multiplyEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
isect_memberEquality,
voidElimination,
addEquality,
voidEquality,
minusEquality,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
intEquality,
universeEquality,
applyLambdaEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
functionEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}m:\mBbbN{}\msupplus{}. ((n \mleq{} m) {}\mRightarrow{} (m \mleq{} (n * |x m|))) supposing 5 \mleq{} |x n|
Date html generated:
2017_10_02-PM-07_16_09
Last ObjectModification:
2017_07_28-AM-07_20_52
Theory : reals
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