Nuprl Lemma : geo-krippen-aux
∀e:BasicGeometry. ∀a1,a2,b1,b2,c,m1,m2:Point.
(a1_c_a2 ⇒ b1_c_b2 ⇒ ca1 ≅ cb1 ⇒ ca2 ≅ cb2 ⇒ a1=m1=b1 ⇒ a2=m2=b2 ⇒ |ca1| ≤ |ca2| ⇒ m1_c_m2)
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
geo-length: |s|,
geo-midpoint: a=m=b,
geo-mk-seg: 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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
stable: Stable{P},
uimplies: b supposing a,
not: ¬A,
subtype_rel: A ⊆r B,
prop: ℙ,
or: P ∨ Q,
exists: ∃x:A. B[x],
false: False,
geo-eq: a ≡ b,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
geo-midpoint: a=m=b,
cand: A c∧ B,
uiff: uiff(P;Q),
geo-out: out(p ab),
squash: ↓T,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
stable__geo-between,
minimal-double-negation-hyp-elim,
false_wf,
or_wf,
geo-sep_wf,
not_wf,
geo-between_wf,
symmetric-point-construction,
geo-sep-sym,
geo-congruent-symmetry,
geo-congruent-sep,
geo-midpoint_functionality,
geo-eq_weakening,
geo-between_functionality,
symmetric-point-unicity,
geo-congruent_functionality,
geo-between-trivial,
minimal-not-not-excluded-middle,
geo-congruence-identity,
geo-midpoint-id2,
geo-le_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-midpoint_wf,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-between-symmetry,
geo-midpoint-diagonals-between,
geo-midpoint-diagonals-congruent2,
geo-congruent-iff-length,
geo-length-flip,
and_wf,
equal_wf,
geo-length-type_wf,
geo-sep-irrefl',
geo-midpoint-id3,
geo-between-trivial2,
geo-sep_functionality,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
geo-out-le-iff-bet,
geo-between-same-side2,
squash_wf,
true_wf,
geo-congruence-identity-sym,
geo-between-outer-trans,
not-not-double-pasch,
exists_wf,
double-negation-hyp-elim,
geo-between-exchange3,
geo-between-inner-trans,
geo-congruent-refl,
geo-inner-five-segment',
at-most-one-midpoint,
stable__geo-congruent,
geo-colinear-between,
geo-colinear-congruence1,
geo-between-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
independent_isectElimination,
applyEquality,
because_Cache,
sqequalRule,
functionEquality,
independent_functionElimination,
unionElimination,
productElimination,
voidElimination,
promote_hyp,
instantiate,
independent_pairFormation,
addLevel,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
applyLambdaEquality,
levelHypothesis,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}a1,a2,b1,b2,c,m1,m2:Point.
(a1\_c\_a2 {}\mRightarrow{} b1\_c\_b2 {}\mRightarrow{} ca1 \00D0 cb1 {}\mRightarrow{} ca2 \00D0 cb2 {}\mRightarrow{} a1=m1=b1 {}\mRightarrow{} a2=m2=b2 {}\mRightarrow{} |ca1| \mleq{} |ca2| {}\mRightarrow{} m1\_c\_m2)
Date html generated:
2017_10_02-PM-06_35_38
Last ObjectModification:
2017_08_17-PM-08_12_20
Theory : euclidean!plane!geometry
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