Nuprl Lemma : eu-congruent-between-exists
∀e:EuclideanPlane. ∀a,b,c,a',c':Point.
(∃b':Point. (a'_b'_c' ∧ ab=a'b' ∧ bc=b'c')) supposing (a_b_c and ac=a'c' and (¬(a = b ∈ Point)))
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,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q)
Lemmas referenced :
eu-point_wf,
eu-between-eq_wf,
eu-congruent_wf,
not_wf,
equal_wf,
euclidean-plane_wf,
eu-congruence-identity,
false_wf,
eu-between-eq-same,
eu-congruence-identity-sym,
eu-extend-exists,
eu-construction-unicity,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-between-eq-exchange4,
eu-three-segment,
eu-congruent-iff-length,
eu-between-eq-outer-trans,
eu-congruence-identity3,
and_wf,
eu-mk-seg_wf,
eu-segment_wf,
eu-length_wf
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,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
because_Cache,
independent_functionElimination,
independent_isectElimination,
equalityTransitivity,
universeEquality,
dependent_set_memberEquality,
productElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality,
applyEquality,
setEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,a',c':Point.
(\mexists{}b':Point. (a'\_b'\_c' \mwedge{} ab=a'b' \mwedge{} bc=b'c')) supposing (a\_b\_c and ac=a'c' and (\mneg{}(a = b)))
Date html generated:
2016_10_26-AM-07_42_31
Last ObjectModification:
2016_07_12-AM-08_09_23
Theory : euclidean!geometry
Home
Index