Nuprl Lemma : geo-congruent_functionality
∀e:EuclideanPlane. ∀a1,a2,b1,b2,c1,c2,d1,d2:Point.
  (a1 ≡ a2 
⇒ b1 ≡ b2 
⇒ c1 ≡ c2 
⇒ d1 ≡ d2 
⇒ (a1b1 ≅ c1d1 
⇐⇒ a2b2 ≅ c2d2))
Proof
Definitions occuring in Statement : 
euclidean-plane: EuclideanPlane
, 
geo-eq: a ≡ b
, 
geo-congruent: ab ≅ cd
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
euclidean-plane: EuclideanPlane
, 
implies: P 
⇒ Q
, 
sq_stable: SqStable(P)
, 
and: P ∧ Q
, 
squash: ↓T
, 
uall: ∀[x:A]. B[x]
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
iff: P 
⇐⇒ Q
, 
guard: {T}
, 
uimplies: b supposing a
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
basic-geo-axioms-imply, 
sq_stable__geo-axioms, 
geo-congruent-functionality-lemma, 
euclidean-plane_wf, 
geo-point_wf, 
euclidean-plane-structure-subtype, 
all_wf, 
geo-eq_wf, 
geo-congruent_wf, 
sq_stable__all, 
sq_stable__geo-congruent, 
euclidean-plane-subtype, 
subtype_rel_transitivity, 
euclidean-plane-structure_wf, 
geo-primitives_wf, 
geo-eq_inversion
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
independent_functionElimination, 
productElimination, 
sqequalRule, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
isectElimination, 
applyEquality, 
lambdaEquality, 
because_Cache, 
functionEquality, 
independent_pairFormation, 
instantiate, 
independent_isectElimination
Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a1,a2,b1,b2,c1,c2,d1,d2:Point.
    (a1  \mequiv{}  a2  {}\mRightarrow{}  b1  \mequiv{}  b2  {}\mRightarrow{}  c1  \mequiv{}  c2  {}\mRightarrow{}  d1  \mequiv{}  d2  {}\mRightarrow{}  (a1b1  \00D0  c1d1  \mLeftarrow{}{}\mRightarrow{}  a2b2  \00D0  c2d2))
Date html generated:
2017_10_02-PM-03_28_45
Last ObjectModification:
2017_08_10-PM-10_52_20
Theory : euclidean!plane!geometry
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