Nuprl Lemma : geo-seg-congruent_functionality
∀e:BasicGeometry. ∀s1,s2,t1,t2:geo-segment(e).
(geo-seg-congruent(e; s1; t1)
⇒ geo-seg-congruent(e; s2; t2)
⇒ (geo-seg-congruent(e; s1; s2) ⇐⇒ geo-seg-congruent(e; t1; t2)))
Proof
Definitions occuring in Statement :
geo-seg-congruent: geo-seg-congruent(e; s1; s2),
geo-segment: geo-segment(e),
basic-geometry: BasicGeometry,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
rev_implies: P ⇐ Q,
uimplies: b supposing a
Lemmas referenced :
geo-seg-congruent_wf,
geo-segment_wf,
basic-geometry_wf,
geo-seg-congruent_transitivity,
geo-seg-congruent_symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
dependent_functionElimination,
independent_isectElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}s1,s2,t1,t2:geo-segment(e).
(geo-seg-congruent(e; s1; t1)
{}\mRightarrow{} geo-seg-congruent(e; s2; t2)
{}\mRightarrow{} (geo-seg-congruent(e; s1; s2) \mLeftarrow{}{}\mRightarrow{} geo-seg-congruent(e; t1; t2)))
Date html generated:
2019_10_16-PM-01_15_03
Last ObjectModification:
2018_09_15-AM-10_16_42
Theory : euclidean!plane!geometry
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