Nuprl Lemma : eu-extend-equal-iff-congruent
∀e:EuclideanPlane
∀[a,b,c,d,c',d':Point]. uiff((extend ab by cd) = (extend ab by c'd') ∈ Point;cd=c'd') supposing ¬(a = b ∈ Point)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-extend: (extend ab by cd),
eu-congruent: ab=cd,
eu-point: Point,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
euclidean-plane: EuclideanPlane,
prop: ℙ,
sq_stable: SqStable(P),
and: P ∧ Q,
uiff: uiff(P;Q),
squash: ↓T,
cand: A c∧ B
Lemmas referenced :
eu-point_wf,
sq_stable__uiff,
equal_wf,
eu-extend_wf,
not_wf,
eu-congruent_wf,
sq_stable__equal,
sq_stable__eu-congruent,
eu-extend-property,
eu-between-eq_wf,
euclidean-plane_wf,
eu-congruent-symmetry,
and_wf,
eu-congruent-transitivity,
eu-congruent-refl,
eu-five-segment,
eu-congruence-identity,
eu-three-segment
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
equalityEquality,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
because_Cache,
dependent_set_memberEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
axiomEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
applyEquality,
setEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[a,b,c,d,c',d':Point]. uiff((extend ab by cd) = (extend ab by c'd');cd=c'd') supposing \mneg{}(a = b)
Date html generated:
2016_10_26-AM-07_41_10
Last ObjectModification:
2016_07_12-AM-08_07_21
Theory : euclidean!geometry
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