Nuprl Lemma : eu-cong-angle-symm
∀e:EuclideanPlane. ∀a,b,c:Point. abc = cba supposing (¬(a = b ∈ Point)) ∧ (¬(c = b ∈ Point))
Proof
Definitions occuring in Statement :
eu-cong-angle: abc = xyz,
euclidean-plane: EuclideanPlane,
eu-point: Point,
uimplies: b supposing a,
all: ∀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,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
eu-cong-angle: abc = xyz,
cand: A c∧ B,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x]
Lemmas referenced :
eu-three-segment,
eu-length-flip,
eu-between-eq-symmetry,
eu-extend-exists,
exists_wf,
eu-congruent_wf,
eu-between-eq_wf,
eu-congruent-flip,
eu-congruent-iff-length,
euclidean-plane_wf,
equal_wf,
not_wf,
eu-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
equalityEquality,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
productEquality,
because_Cache,
independent_pairFormation,
independent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality,
independent_isectElimination,
dependent_pairFormation,
equalityTransitivity
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. abc = cba supposing (\mneg{}(a = b)) \mwedge{} (\mneg{}(c = b))
Date html generated:
2016_06_16-PM-01_32_01
Last ObjectModification:
2016_05_23-AM-11_11_03
Theory : euclidean!geometry
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