Nuprl Lemma : geo-cong-preserves-bet-out
∀e:BasicGeometry. ∀a,b,c,a',b',c':Point. (a_b_c ⇒ ab ≅ a'b' ⇒ ac ≅ a'c' ⇒ out(a' b'c') ⇒ a'_b'_c')
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
guard: {T},
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
uimplies: b supposing a,
stable: Stable{P},
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
member: t ∈ T,
and: P ∧ Q,
geo-out: out(p ab),
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-out_wf,
not_wf,
geo-between_wf,
stable__geo-between,
geo-add-length-between,
geo-add-length_wf,
geo-mk-seg_wf,
geo-length_wf,
iff_weakening_equal,
geo-length-type_wf,
true_wf,
squash_wf,
equal_wf,
geo-congruent-iff-length,
geo-add-length-implies-eq-zero,
geo-add-length-assoc,
geo-add-length-is-zero,
geo-zero-length-iff,
geo-eq_weakening,
geo-between_functionality,
geo-between-trivial
Rules used in proof :
instantiate,
voidElimination,
sqequalRule,
because_Cache,
applyEquality,
independent_pairFormation,
independent_functionElimination,
independent_isectElimination,
isectElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',b',c':Point.
(a\_b\_c {}\mRightarrow{} ab \00D0 a'b' {}\mRightarrow{} ac \00D0 a'c' {}\mRightarrow{} out(a' b'c') {}\mRightarrow{} a'\_b'\_c')
Date html generated:
2017_10_02-PM-06_27_54
Last ObjectModification:
2017_08_05-PM-04_40_57
Theory : euclidean!plane!geometry
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