Nuprl Lemma : Euclid-Prop25
∀p:EuclideanPlane. ∀a,b,c,d,e,f:Point. (a # bc ⇒ d # ef ⇒ ab ≅ de ⇒ ac ≅ df ⇒ |ef| < |bc| ⇒ edf < bac)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-congruent: ab ≅ cd,
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,
guard: {T},
and: P ∧ Q,
or: P ∨ Q,
geo-lsep: a # bc,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cand: A c∧ B,
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
oriented-plane: OrientedPlane,
sq_stable: SqStable(P),
squash: ↓T,
geo-midpoint: a=m=b,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-lt-angle: abc < xyz,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
true: True,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
geo-out: out(p ab),
geo-eq: a ≡ b,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c
Lemmas referenced :
geo-lt-implies-point,
lsep-implies-sep,
lsep-all-sym,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lsep_wf,
geo-point_wf,
Euclid-Prop23_half-plane2,
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
left-implies-sep,
geo-sep-O-X,
geo-sep-sym,
geo-strict-between-sep3,
geo-out-if-between,
geo-strict-between-sym,
geo-left-out-1,
geo-out_inversion,
geo-left-out-3,
geo-out_weakening,
geo-eq_weakening,
geo-cong-angle-symmetry,
geo-left_wf,
geo-cong-angle_wf,
out-preserves-angle-cong_1,
geo-left-out-2,
geo-between-out,
geo-strict-between-sep1,
geo-strict-between-implies-between,
geo-sas2,
geo-congruent-iff-length,
isosceles-sep-implies-lsep,
geo-midpoint_wf,
geo-sep_wf,
left-between,
sq_stable__geo-between,
geo-between-symmetry,
colinear-lsep-cycle,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
geo-sep-or,
use-plane-sep_strict,
left-symmetry,
lsep-iff-all-sep,
geo-out-colinear,
lsep-all-sym2,
not-lsep-if-colinear,
geo-out_wf,
geo-cong-angle-refl,
geo-between-trivial,
lsep-not-between,
geo-colinear-append,
cons_wf,
nil_wf,
length_wf,
select_wf,
nat_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
l_member_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
euclidean-plane-axioms,
geo-strict-between-sep2,
geo-between_wf,
Euclid-Prop24,
geo-cong-angle-preserves-lt-angle,
geo-colinear-cases,
false_wf,
stable__false,
geo-eq_wf,
geo-between-sep,
geo-strict-between_wf,
not-left-and-right,
left-between-implies-right1,
left-convex2,
not-gt-and-lt,
out-cong-angle,
geo-cong-angle-symm2,
lsep-symmetry,
geo-cong-angle-preserves-lt-angle2,
Euclid-Prop23_half-plane,
geo-left-out,
left-between-implies-right2,
left-convex,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
productElimination,
unionElimination,
universeIsType,
isectElimination,
sqequalRule,
setElimination,
rename,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
equalitySymmetry,
equalityTransitivity,
setIsType,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
lambdaEquality_alt,
equalityIstype,
int_eqEquality,
functionIsType,
inrFormation_alt
Latex:
\mforall{}p:EuclideanPlane. \mforall{}a,b,c,d,e,f:Point.
(a \# bc {}\mRightarrow{} d \# ef {}\mRightarrow{} ab \mcong{} de {}\mRightarrow{} ac \mcong{} df {}\mRightarrow{} |ef| < |bc| {}\mRightarrow{} edf < bac)
Date html generated:
2019_10_16-PM-02_35_16
Last ObjectModification:
2019_09_24-PM-02_06_33
Theory : euclidean!plane!geometry
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