Nuprl Lemma : rng-pp-nontrivial-1-ext
∀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 :
uimplies: b supposing a,
so_apply: x[s],
top: Top,
so_lambda: λ2x.t[x],
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],
decidable__int_equal,
any: any x,
decidable__equal_int,
rng-pp-nontrivial-1,
ifthenelse: if b then t else f fi ,
bfalse: ff,
it: ⋅,
btrue: tt,
eq_int: (i =z j),
member: t ∈ T
Lemmas referenced :
strict4-decide,
lifting-strict-int_eq,
rng-pp-nontrivial-1,
decidable__int_equal,
decidable__equal_int
Rules used in proof :
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{}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:
2018_05_22-PM-00_53_34
Last ObjectModification:
2018_05_21-AM-01_22_57
Theory : euclidean!plane!geometry
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