Nuprl Lemma : rng-pp-nontrivial-1
∀r:IntegDom{i}
((∀x,y:|r|. Dec(x = y ∈ |r|))
⇒ (∀l:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))}
∃a:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))}
(∃b:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} [(((a . l) = 0 ∈ |r|)
∧ ((b . l) = 0 ∈ |r|)
∧ (¬((a x b) = 0 ∈ (ℕ3 ⟶ |r|))))])))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
decidable: Dec(P),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
integ_dom: IntegDom{i},
rng_zero: 0,
rng_car: |r|,
scalar-product: (a . b),
cross-product: (a x b),
zero-vector: 0
Definitions unfolded in proof :
int_seg: {i..j-},
uimplies: b supposing a,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
rng: Rng,
crng: CRng,
integ_dom: IntegDom{i},
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
bfalse: ff,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
sq_type: SQType(T),
or: P ∨ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
guard: {T},
zero-vector: 0,
subtype_rel: A ⊆r B,
squash: ↓T,
less_than: a < b,
infix_ap: x f y,
true: True,
eq_int: (i =z j),
decidable: Dec(P),
sq_exists: ∃x:A [B[x]],
cross-product: (a x b),
select: L[n],
cons: [a / b],
integ_dom_p: IsIntegDom(r),
nequal: a ≠ b ∈ T ,
subtract: n - m
Lemmas referenced :
integ_dom_wf,
decidable_wf,
all_wf,
int-value-type,
lelt_wf,
set-value-type,
zero-vector_wf,
le_wf,
false_wf,
non-zero-component_wf,
rng_zero_wf,
rng_car_wf,
equal_wf,
not_wf,
int_seg_wf,
set_wf,
scalar-product_wf,
rng_minus_wf,
eq_int_wf,
ifthenelse_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
int_seg_properties,
member_wf,
bnot_wf,
assert_wf,
rng_plus_comm,
rng_plus_inv_assoc,
crng_times_comm,
rng_times_over_minus,
iff_weakening_equal,
rng_plus_zero,
rng_plus_ac_1,
rng_plus_assoc,
subtype_rel_self,
rng_times_zero,
true_wf,
squash_wf,
rng_times_wf,
infix_ap_wf,
rng_plus_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
intformless_wf,
intformand_wf,
int_seg_subtype,
int_seg_cases,
int_subtype_base,
decidable__equal_int,
scalar-product-3,
decidable__le,
istype-int,
istype-void,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_sets_simple,
cross-product_wf,
select_wf,
cons_wf,
nil_wf,
length_of_cons_lemma,
length_of_nil_lemma,
itermAdd_wf,
int_term_value_add_lemma,
istype-universe,
rng_minus_zero,
rng_minus_over_plus,
rng_minus_minus
Rules used in proof :
intEquality,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
cutEval,
functionEquality,
independent_pairFormation,
dependent_set_memberEquality,
hypothesisEquality,
applyEquality,
because_Cache,
setElimination,
lambdaEquality,
sqequalRule,
hypothesis,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
rename,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionExtensionality,
productEquality,
impliesFunctionality,
cumulativity,
instantiate,
unionElimination,
voidEquality,
isect_memberEquality,
dependent_functionElimination,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
productElimination,
voidElimination,
independent_functionElimination,
applyLambdaEquality,
universeEquality,
imageElimination,
baseClosed,
imageMemberEquality,
addEquality,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
universeIsType,
productIsType,
lambdaFormation_alt,
equalityIstype,
sqequalBase,
functionIsType,
inhabitedIsType,
dependent_set_memberFormation_alt,
setIsType
Latex:
\mforall{}r:IntegDom\{i\}
((\mforall{}x,y:|r|. Dec(x = y))
{}\mRightarrow{} (\mforall{}l:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\}
\mexists{}a:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} . (\mexists{}b:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} [(((a . l) = 0) \mwedge{} ((b . l) = 0) \mwedge{} \000C(\mneg{}((a x b) = 0)))])))
Date html generated:
2019_10_16-PM-02_13_33
Last ObjectModification:
2019_08_29-PM-02_57_20
Theory : euclidean!plane!geometry
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