Nuprl Lemma : eu-congruent-preserves-between
∀e:EuclideanPlane. ∀[a,b,c,a',b',c':Point]. (a'_b'_c') supposing (bc=b'c' and ac=a'c' and ab=a'b' and a_b_c)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-congruent: ab=cd,
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
stable: Stable{P},
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
false: False,
exists: ∃x:A. B[x],
and: P ∧ Q,
uiff: uiff(P;Q)
Lemmas referenced :
stable__eu-between-eq,
not_wf,
eu-between-eq_wf,
eu-congruent_wf,
eu-point_wf,
euclidean-plane_wf,
equal_wf,
eu-congruence-identity-sym,
eu-between-eq-trivial-left,
eu-congruent-between-exists,
eu-congruent-refl,
eu-congruent-iff-length,
eu-length-flip,
eu-inner-five-segment
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
independent_isectElimination,
because_Cache,
independent_functionElimination,
voidElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
productElimination,
equalityTransitivity
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[a,b,c,a',b',c':Point]. (a'\_b'\_c') supposing (bc=b'c' and ac=a'c' and ab=a'b' and a\_b\_c)
Date html generated:
2016_10_26-AM-07_42_36
Last ObjectModification:
2016_07_12-AM-08_08_56
Theory : euclidean!geometry
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