Nuprl Lemma : rng-pp-nontrivial-3-ext
∀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 :
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
rng-pp-nontrivial-1-ext,
cross-product-non-zero-implies-ext,
any: any x,
deq-implies,
rng-pp-nontrivial-2,
rng-pp-nontrivial-3,
rng-pp-nontriv1: rng-pp-nontriv1(r;eq;l),
it: ⋅,
member: t ∈ T
Lemmas referenced :
is-exception_wf,
has-value_wf_base,
equal_wf,
top_wf,
lifting-strict-int_eq,
lifting-strict-callbyvalue,
strict4-spread,
lifting-strict-spread,
rng-pp-nontrivial-3,
rng-pp-nontrivial-1-ext,
cross-product-non-zero-implies-ext,
deq-implies,
rng-pp-nontrivial-2
Rules used in proof :
closedConclusion,
baseApply,
exceptionSqequal,
axiomSqleEquality,
spreadExceptionCases,
independent_functionElimination,
dependent_functionElimination,
hypothesisEquality,
sqleReflexivity,
productElimination,
productEquality,
callbyvalueSpread,
divergentSqle,
sqequalSqle,
lambdaFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
equalitySymmetry,
equalityTransitivity,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
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_54_18
Last ObjectModification:
2018_05_21-AM-01_24_59
Theory : euclidean!plane!geometry
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