Nuprl Lemma : geo-Aparallel_functionality
∀[e1:EuclideanPlane]. ∀[l,m,l2,m2:Line]. ({uiff(l || m;l2 || m2)}) supposing (m ≡ m2 and l ≡ l2)
Proof
Definitions occuring in Statement :
geo-Aparallel: l || m,
geo-line-eq: l ≡ m,
geo-line: Line,
euclidean-plane: EuclideanPlane,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
geoline: LINE,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_apply: x[s1;s2],
implies: P ⇒ Q,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
geo-Aparallel: l || m,
not: ¬A,
false: False,
prop: ℙ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
quotient-member-eq,
geo-line-eq_wf,
geo-line_wf,
geo-line-eq-equiv,
geo-intersect_wf,
geoline-subtype1,
geo-Aparallel_wf,
squash_wf,
true_wf,
geoline_wf,
subtype_rel_self,
iff_weakening_equal,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality,
hypothesisEquality,
applyEquality,
hypothesis,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
voidElimination,
addLevel,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination,
cumulativity,
isectEquality,
independent_pairEquality,
isect_memberEquality
Latex:
\mforall{}[e1:EuclideanPlane]. \mforall{}[l,m,l2,m2:Line]. (\{uiff(l || m;l2 || m2)\}) supposing (m \mequiv{} m2 and l \mequiv{} l2)
Date html generated:
2018_05_22-PM-01_09_39
Last ObjectModification:
2018_05_11-PM-02_17_58
Theory : euclidean!plane!geometry
Home
Index