Nuprl Lemma : geo-Aparallel_weakening
∀g:EuclideanPlane. ∀l,m:LINE. ((l = m ∈ LINE) ⇒ l || m)
Proof
Definitions occuring in Statement :
geo-Aparallel: l || m,
geoline: LINE,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
geo-Aparallel: l || m
Lemmas referenced :
geo-Aparallel_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
geo-intersect-irreflexive,
equal_wf,
geoline_wf,
euclidean-plane_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}l,m:LINE. ((l = m) {}\mRightarrow{} l || m)
Date html generated:
2018_05_22-PM-01_10_21
Last ObjectModification:
2018_05_10-PM-06_36_45
Theory : euclidean!plane!geometry
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