Nuprl Lemma : rmul-is-negative1
∀x,y:ℝ. (((x * y) < r0) ⇒ (x ≠ r0 ∨ y ≠ r0))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rless: x < y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rneq: x ≠ y,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
int-to-real: r(n),
rmul: a * b,
has-value: (a)↓,
exists: ∃x:A. B[x],
reg-seq-mul: reg-seq-mul(x;y),
accelerate: accelerate(k;f),
uimplies: b supposing a,
real: ℝ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
subtype_rel: A ⊆r B,
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
int_upper: {i...},
guard: {T},
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
true: True,
int_nzero: ℤ-o,
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
less_than': less_than'(a;b),
absval: |i|,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rless_wf,
rmul_wf,
int-to-real_wf,
real_wf,
real-has-value,
value-type-has-value,
int-value-type,
imax_wf,
absval_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rless-iff2,
false_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformand_wf,
add-is-int-iff,
int_upper_properties,
int_entire_a,
nequal_wf,
int_subtype_base,
subtype_base_sq,
mul_nzero,
mul_nat_plus,
istype-le,
decidable__le,
nat_properties,
imax_nat,
int_seg_properties,
int_seg_cases,
int_seg_subtype_special,
decidable__equal_int,
div_is_zero,
nat_plus_subtype_nat,
mul_preserves_le,
zero-div-rem,
int_term_value_minus_lemma,
itermMinus_wf,
absval_ubound,
div_absval_bound
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
inhabitedIsType,
sqequalRule,
callbyvalueReduce,
productElimination,
intEquality,
independent_isectElimination,
multiplyEquality,
addEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
unionIsType,
inlFormation_alt,
inrFormation_alt,
independent_pairFormation,
int_eqEquality,
baseApply,
promote_hyp,
pointwiseFunctionality,
applyLambdaEquality,
sqequalBase,
baseClosed,
equalityIstype,
cumulativity,
instantiate,
closedConclusion,
divideEquality,
minusEquality,
productIsType,
imageElimination,
hypothesis_subsumption
Latex:
\mforall{}x,y:\mBbbR{}. (((x * y) < r0) {}\mRightarrow{} (x \mneq{} r0 \mvee{} y \mneq{} r0))
Date html generated:
2019_10_29-AM-10_05_18
Last ObjectModification:
2019_04_01-PM-11_22_07
Theory : reals
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