Nuprl Lemma : geo-SCS-congruent
∀g:EuclideanPlane. ∀c,d,a:Point. ∀b:{b:Point| b ≠ a ∧ c_b_d} . cSCS(a;b;c;d) ≅ cd
Proof
Definitions occuring in Statement :
geo-SCS: SCS(a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
euclidean-plane: EuclideanPlane,
guard: {T},
uimplies: b supposing a,
implies: P ⇒ Q,
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
geo-SCS_wf,
set_wf,
geo-point_wf,
geo-congruent_wf,
geo-between_wf,
geo-SCO_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-colinear_wf,
sq_stable__geo-congruent,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
setElimination,
rename,
setEquality,
instantiate,
independent_isectElimination,
functionEquality,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}g:EuclideanPlane. \mforall{}c,d,a:Point. \mforall{}b:\{b:Point| b \mneq{} a \mwedge{} c\_b\_d\} . cSCS(a;b;c;d) \mcong{} cd
Date html generated:
2018_05_22-AM-11_55_03
Last ObjectModification:
2018_03_30-PM-06_09_10
Theory : euclidean!plane!geometry
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