Nuprl Lemma : geo-perp-is-shortest-path
∀e:EuclideanPlane. ∀a,b,c,d:Point. (a # bc ⇒ ad ⊥d bc ⇒ (∀x:{x:Point| Colinear(b;c;x)} . (d ≠ x ⇒ |ad| < |ax|)))
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
guard: {T},
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
or: P ∨ Q,
uimplies: b supposing a,
basic-geometry: BasicGeometry,
cand: A c∧ B,
geo-perp-in: ab ⊥x cd,
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),
exists: ∃x:A. B[x],
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
sq_stable: SqStable(P),
l_member: (x ∈ l),
nat: ℕ,
less_than: a < b,
squash: ↓T,
less_than': 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],
basic-geometry-: BasicGeometry-,
geo-strict-between: a-b-c,
geo-midpoint: a=m=b,
uiff: uiff(P;Q),
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
geo-sep-or,
lsep-implies-sep,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-perp-in_wf,
geo-lsep_wf,
geo-point_wf,
colinear-lsep,
lsep-all-sym,
geo-sep-sym,
geo-colinear-is-colinear-set,
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,
sq_stable__colinear,
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,
Euclid-Prop20,
colinear-lsep-cycle,
geo-strict-between-sep1,
geo-strict-between-implies-colinear,
geo-add-length-between,
geo-proper-extend-exists,
geo-reflected-right-triangles-congruent,
geo-colinear-same,
right-angle-symmetry,
geo-congruent-iff-length,
geo-length-flip,
geo-add-length_wf,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length-comm,
geo-add-length-lt-cancel-for-double,
geo-length_wf1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
independent_functionElimination,
productElimination,
dependent_set_memberEquality_alt,
universeIsType,
isectElimination,
applyEquality,
sqequalRule,
unionElimination,
instantiate,
independent_isectElimination,
setIsType,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
independent_pairFormation,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
imageMemberEquality,
baseClosed,
equalityIstype,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
(a \# bc {}\mRightarrow{} ad \mbot{}d bc {}\mRightarrow{} (\mforall{}x:\{x:Point| Colinear(b;c;x)\} . (d \mneq{} x {}\mRightarrow{} |ad| < |ax|)))
Date html generated:
2019_10_16-PM-02_21_01
Last ObjectModification:
2019_03_20-PM-03_02_22
Theory : euclidean!plane!geometry
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