Nuprl Lemma : geo-out-cong-cong
∀e:BasicGeometry. ∀a,b,c,a',b',c':Point. (out(a bc) ⇒ out(a' b'c') ⇒ ab ≅ a'b' ⇒ ac ≅ a'c' ⇒ bc ≅ b'c')
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
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],
or: P ∨ Q,
not: ¬A,
stable: Stable{P},
basic-geometry: BasicGeometry,
false: False,
and: P ∧ Q,
uiff: uiff(P;Q),
geo-out: out(p ab)
Lemmas referenced :
geo-point_wf,
geo-out_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,
minimal-double-negation-hyp-elim,
not_wf,
geo-between_wf,
or_wf,
false_wf,
stable__geo-congruent,
geo-cong-preserves-bet-out,
geo-add-length-cancel-left,
geo-mk-seg_wf,
geo-length_wf,
geo-add-length_wf,
geo-length-type_wf,
equal_wf,
and_wf,
geo-congruent-iff-length,
geo-add-length-between,
geo-out_inversion,
geo-length-flip
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,
unionElimination,
independent_functionElimination,
functionEquality,
rename,
setElimination,
dependent_functionElimination,
voidElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
dependent_set_memberEquality,
productElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',b',c':Point.
(out(a bc) {}\mRightarrow{} out(a' b'c') {}\mRightarrow{} ab \00D0 a'b' {}\mRightarrow{} ac \00D0 a'c' {}\mRightarrow{} bc \00D0 b'c')
Date html generated:
2017_10_02-PM-06_28_01
Last ObjectModification:
2017_08_05-PM-04_41_02
Theory : euclidean!plane!geometry
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