Nuprl Lemma : geo-strict-between-congruence
∀e:BasicGeometry. ∀A,B,C,C':Point. (A_C_B ⇒ AC ≅ AC' ⇒ BC ≅ BC' ⇒ C ≡ C')
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
false: False,
or: P ∨ Q,
not: ¬A,
stable: Stable{P},
geo-eq: a ≡ b
Lemmas referenced :
geo-point_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
minimal-not-not-excluded-middle,
geo-congruence-identity,
geo-between-same,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality,
minimal-double-negation-hyp-elim,
not_wf,
or_wf,
false_wf,
geo-sep_wf,
stable__not,
geo-between-implies-colinear,
geo-colinear-permute,
geo-colinear-cycle,
geo-colinear-congruence2
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
promote_hyp,
productElimination,
dependent_functionElimination,
voidElimination,
unionElimination,
independent_functionElimination,
functionEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C,C':Point. (A\_C\_B {}\mRightarrow{} AC \00D0 AC' {}\mRightarrow{} BC \00D0 BC' {}\mRightarrow{} C \mequiv{} C')
Date html generated:
2017_10_02-PM-06_32_04
Last ObjectModification:
2017_08_05-PM-04_42_56
Theory : euclidean!plane!geometry
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