Nuprl Lemma : cong-angle-out-aux2_1
∀g:BasicGeometry. ∀a,b,c,d,e,f,a',c',d',f':Point.
(a'c' ≅ d'f' ⇒ out(b a'a) ⇒ out(b c'c) ⇒ out(e d'd) ⇒ out(e f'f) ⇒ ba' ≅ ed' ⇒ bc' ≅ ef' ⇒ abc ≅a def)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
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,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
basic-geometry: BasicGeometry,
geo-out: out(p ab),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
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,
geo-eq: a ≡ b
Lemmas referenced :
geo-sep-sym,
geo-proper-extend-exists,
geo-strict-between-implies-between,
subtype_rel_self,
basic-geometry-_wf,
geo-between-symmetry,
geo-congruent-iff-length,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
geo-add-length-comm,
geo-between_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-out_wf,
geo-point_wf,
oriented-colinear-append,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-sep_wf,
geo-colinear-is-colinear-set,
geo-out-colinear,
geo-strict-between-implies-colinear,
list_ind_cons_lemma,
istype-void,
list_ind_nil_lemma,
length_of_cons_lemma,
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-colinear-cases,
stable__geo-congruent,
geo-eq_wf,
geo-strict-between_wf,
geo-between-out-implies-out,
geo-out_inversion,
or_comm,
geo-cong-preserves-bet-out,
geo-be-end-eq,
geo-between-trivial,
geo-between_functionality,
geo-eq_weakening,
geo-colinear_functionality,
geo-congruent_functionality,
geo-congruent-refl,
geo-strict-between-sep1,
geo-cong-preserves-strict-bet-out,
geo-inner-three-segment,
geo-length-flip,
geo-five-segment,
geo-strict-between-sym,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4,
geo-between-outer-trans,
geo-strict-between-sep3,
geo-inner-five-segment,
geo-not-bet-and-out,
geo-out_transitivity,
geo-between-out,
geo-between-implies-colinear
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
hypothesis,
because_Cache,
rename,
dependent_pairFormation_alt,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productIsType,
inlFormation_alt,
inrFormation_alt,
equalityIstype,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
setElimination,
functionIsType
Latex:
\mforall{}g:BasicGeometry. \mforall{}a,b,c,d,e,f,a',c',d',f':Point.
(a'c' \mcong{} d'f'
{}\mRightarrow{} out(b a'a)
{}\mRightarrow{} out(b c'c)
{}\mRightarrow{} out(e d'd)
{}\mRightarrow{} out(e f'f)
{}\mRightarrow{} ba' \mcong{} ed'
{}\mRightarrow{} bc' \mcong{} ef'
{}\mRightarrow{} abc \mcong{}\msuba{} def)
Date html generated:
2019_10_16-PM-01_26_10
Last ObjectModification:
2018_12_15-PM-10_01_32
Theory : euclidean!plane!geometry
Home
Index