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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