Nuprl Lemma : hp-angle-sum-lt3
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(a' # b'c'
⇒ abc + xyz ≅ ijk
⇒ a'b'c' + x'y'z' ≅ i'j'k'
⇒ ijk ≅a i'j'k'
⇒ a # bc
⇒ x # yz
⇒ i # jk
⇒ abc < a'b'c'
⇒ x'y'z' < xyz)
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-lt-angle: abc < xyz,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
hp-angle-sum: abc + xyz ≅ def,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-lt-angle: abc < xyz,
geo-out: out(p ab),
basic-geometry-: BasicGeometry-,
cand: A c∧ 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),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-cong-angle: abc ≅a xyz,
uiff: uiff(P;Q),
geo-tri: Triangle(a;b;c),
squash: ↓T,
true: True,
geo-strict-between: a-b-c,
geo-lsep: a # bc,
oriented-plane: OrientedPlane,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
geo-point_wf,
hp-angle-sum_wf,
geo-cong-angle_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-lsep_wf,
geo-lt-angle_wf,
geo-cong-angle-preserves-lt-angle2,
geo-cong-angle-symm2,
cong-angle-preserves-lsep_strong,
geo-proper-extend-exists,
geo-sep-sym,
geo-strict-between-sep3,
geo-out-if-between,
geo-strict-between-sym,
out-preserves-lsep,
lsep-symmetry,
lsep-all-sym,
colinear-lsep-cycle,
geo-strict-between-sep2,
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,
lsep-implies-sep,
geo-out_inversion,
euclidean-plane-axioms,
geo-sas,
geo-congruent-iff-length,
geo-cong-angle-transitivity,
geo-between-sep,
out-preserves-angle-cong_1,
geo-between_wf,
geo-strict-between-sep1,
geo-five-segment,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-length-flip,
geo-between-trivial,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-congruent_wf,
geo-out_weakening,
geo-eq_weakening,
geo-out_transitivity,
geo-between-implies-colinear,
geo-between-out,
left-symmetry,
geo-left-out-4,
between-preserves-left-2,
between-preserves-left-1,
geo-left_wf,
between-preserves-left-4,
between-preserves-left-3,
unique-angles-in-half-plane-better2,
geo-cong-angle-symmetry,
geo-cong-angle-symm3,
unique-angles-in-half-plane-better,
geo-out-interior-point-exists,
geo-out-colinear,
not-out-if-lsep,
lsep-not-between,
geo-out_wf,
geo-sep_wf,
out-cong-angle,
lt-angle-implies-between-if-out,
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,
geo-between-lt-angle,
geo-strict-between-trans,
geo-cong-angle-preserves-lt-angle,
cong-tri-implies-cong-angle2
Rules used in proof :
inhabitedIsType,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
universeIsType,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
rename,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
equalitySymmetry,
functionIsType,
equalityTransitivity,
imageElimination,
imageMemberEquality,
baseClosed,
setElimination,
equalityIstype,
int_eqEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(a' \# b'c'
{}\mRightarrow{} abc + xyz \mcong{} ijk
{}\mRightarrow{} a'b'c' + x'y'z' \mcong{} i'j'k'
{}\mRightarrow{} ijk \mcong{}\msuba{} i'j'k'
{}\mRightarrow{} a \# bc
{}\mRightarrow{} x \# yz
{}\mRightarrow{} i \# jk
{}\mRightarrow{} abc < a'b'c'
{}\mRightarrow{} x'y'z' < xyz)
Date html generated:
2019_10_16-PM-02_24_22
Last ObjectModification:
2019_10_02-AM-10_31_31
Theory : euclidean!plane!geometry
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