Nuprl Lemma : rng-pp-nontriv2_wf
∀[r:IntegDom{i}]. ∀[eq:EqDecider(|r|)]. ∀[a:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} ].
(rng-pp-nontriv2(r;eq;a) ∈ ∃l,m,n:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))}
((¬¬((a . l) = 0 ∈ |r|))
∧ (¬¬((a . m) = 0 ∈ |r|))
∧ (¬¬((a . n) = 0 ∈ |r|))
∧ (∃a:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))}
((¬¬((a . l) = 0 ∈ |r|)) ∧ (¬((a . m) = 0 ∈ |r|))))
∧ (∃a:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))}
((¬¬((a . m) = 0 ∈ |r|)) ∧ (¬((a . n) = 0 ∈ |r|))))
∧ (∃a:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))}
((¬¬((a . n) = 0 ∈ |r|)) ∧ (¬((a . l) = 0 ∈ |r|))))))
Proof
Definitions occuring in Statement :
rng-pp-nontriv2: rng-pp-nontriv2(r;eq;p),
deq: EqDecider(T),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
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),
zero-vector: 0
Definitions unfolded in proof :
rng-pp-nontrivial-4-ext,
uimplies: b supposing a,
exists: ∃x:A. B[x],
so_apply: x[s],
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
rng: Rng,
crng: CRng,
integ_dom: IntegDom{i},
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set_wf,
subtype_rel_function,
rng_zero_wf,
le_wf,
false_wf,
scalar-product_wf,
exists_wf,
zero-vector_wf,
equal_wf,
not_wf,
int_seg_wf,
rng_car_wf,
deq_wf,
all_wf,
integ_dom_wf,
subtype_rel_self,
rng-pp-nontrivial-4-ext
Rules used in proof :
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_isectElimination,
universeEquality,
cumulativity,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
lambdaFormation,
hypothesisEquality,
functionExtensionality,
natural_numberEquality,
setEquality,
lambdaEquality,
because_Cache,
rename,
setElimination,
functionEquality,
isectElimination,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:IntegDom\{i\}]. \mforall{}[eq:EqDecider(|r|)]. \mforall{}[a:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} ].
(rng-pp-nontriv2(r;eq;a) \mmember{} \mexists{}l,m,n:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\}
((\mneg{}\mneg{}((a . l) = 0))
\mwedge{} (\mneg{}\mneg{}((a . m) = 0))
\mwedge{} (\mneg{}\mneg{}((a . n) = 0))
\mwedge{} (\mexists{}a:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} . ((\mneg{}\mneg{}((a . l) = 0)) \mwedge{} (\mneg{}((a . m) = 0)))\000C)
\mwedge{} (\mexists{}a:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} . ((\mneg{}\mneg{}((a . m) = 0)) \mwedge{} (\mneg{}((a . n) = 0)))\000C)
\mwedge{} (\mexists{}a:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\}
((\mneg{}\mneg{}((a . n) = 0)) \mwedge{} (\mneg{}((a . l) = 0))))))
Date html generated:
2018_05_22-PM-00_55_13
Last ObjectModification:
2018_05_21-AM-01_27_01
Theory : euclidean!plane!geometry
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