Nuprl Lemma : rng-pp-nontrivial-3
∀r:IntegDom{i}. ∀eq:EqDecider(|r|). ∀l:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} .
∃a,b,c:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))}
((¬¬((a . l) = 0 ∈ |r|))
∧ (¬¬((b . l) = 0 ∈ |r|))
∧ (¬¬((c . l) = 0 ∈ |r|))
∧ (∃l:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} . ((¬¬((a . l) = 0 ∈ |r|)) ∧ (¬((b . l) = 0 ∈ |r|))))
∧ (∃l:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} . ((¬¬((b . l) = 0 ∈ |r|)) ∧ (¬((c . l) = 0 ∈ |r|))))
∧ (∃l:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} . ((¬¬((c . l) = 0 ∈ |r|)) ∧ (¬((a . l) = 0 ∈ |r|)))))
Proof
Definitions occuring in Statement :
deq: EqDecider(T),
int_seg: {i..j-},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
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),
zero-vector: 0
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
cand: A c∧ B,
sq_exists: ∃x:A [B[x]],
prop: ℙ,
false: False,
not: ¬A,
and: P ∧ Q,
exists: ∃x:A. B[x],
guard: {T},
implies: P ⇒ Q,
rng: Rng,
crng: CRng,
integ_dom: IntegDom{i},
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
integ_dom_wf,
deq_wf,
set_wf,
rng_zero_wf,
exists_wf,
le_wf,
false_wf,
scalar-product_wf,
cross-product_wf,
zero-vector_wf,
int_seg_wf,
cross-product-non-zero-implies-ext,
equal_wf,
not_wf,
rng-pp-nontrivial-2,
rng_car_wf,
deq-implies
Rules used in proof :
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
sqequalRule,
natural_numberEquality,
functionEquality,
productEquality,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
voidElimination,
promote_hyp,
independent_pairFormation,
dependent_pairFormation,
productElimination,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}r:IntegDom\{i\}. \mforall{}eq:EqDecider(|r|). \mforall{}l:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} .
\mexists{}a,b,c:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\}
((\mneg{}\mneg{}((a . l) = 0))
\mwedge{} (\mneg{}\mneg{}((b . l) = 0))
\mwedge{} (\mneg{}\mneg{}((c . l) = 0))
\mwedge{} (\mexists{}l:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} . ((\mneg{}\mneg{}((a . l) = 0)) \mwedge{} (\mneg{}((b . l) = 0))))
\mwedge{} (\mexists{}l:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} . ((\mneg{}\mneg{}((b . l) = 0)) \mwedge{} (\mneg{}((c . l) = 0))))
\mwedge{} (\mexists{}l:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} . ((\mneg{}\mneg{}((c . l) = 0)) \mwedge{} (\mneg{}((a . l) = 0)))))
Date html generated:
2018_05_22-PM-00_53_56
Last ObjectModification:
2018_05_21-AM-01_24_36
Theory : euclidean!plane!geometry
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