Nuprl Lemma : opp-side_half-plane-angle-congruence
∀e:EuclideanPlane. ∀b,p,b',p',a,c,a',c':Point.
((a leftof bp ∧ c leftof pb) ⇒ (a' leftof b'p' ∧ c' leftof p'b') ⇒ (abp ≅a a'b'p' ∧ pbc ≅a p'b'c') ⇒ abc ≅a a'b'c')
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
geo-eq: a ≡ b,
uiff: uiff(P;Q),
oriented-plane: OrientedPlane,
basic-geometry-: BasicGeometry-,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
ge: i ≥ j ,
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l),
euclidean-plane: EuclideanPlane,
subtract: n - m,
cons: [a / b],
select: L[n],
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
geo-lsep: a # bc,
iff: P ⇐⇒ Q,
or: P ∨ Q,
cand: A c∧ B,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-between-outer-trans,
geo-add-length-comm,
geo-strict-between-sep1,
geo-strict-between-implies-colinear,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4,
geo-out_transitivity,
geo-out_inversion,
geo-congruent-symmetry,
geo-inner-five-segment,
geo-congruent-comm,
geo-congruent-between-exists,
geo-strict-between-sep2,
oriented-colinear-append,
cons_member,
lsep-all-sym2,
euclidean-plane-axioms,
geo-congruent-full-symmetry,
left-between-implies-right1,
geo-left-out,
geo-congruent-sep,
geo-strict-between_functionality,
basic-geometry_wf,
geo-length-type_wf,
true_wf,
squash_wf,
geo-add-length_wf,
geo-add-length-between,
geo-length-flip,
geo-five-segment,
geo-eq_inversion,
geo-strict-between-sep3,
geo-strict-between-implies-between,
geo-between-symmetry,
left-between-implies-right2,
geo-proper-extend-exists,
geo-between_functionality,
geo-congruent_functionality,
geo-left_functionality,
geo-cong-angle_functionality,
geo-lsep_functionality,
geo-sep_functionality,
geo-eq_weakening,
geo-colinear_functionality,
geo-congruent-iff-length,
Euclid-Prop7,
geo-lsep_wf,
geo-colinear_wf,
geo-strict-between_wf,
geo-eq_wf,
stable__geo-congruent,
geo-colinear-cases,
geo-sas2,
list_ind_nil_lemma,
list_ind_cons_lemma,
l_member_wf,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformand_wf,
nat_properties,
select_wf,
length_wf,
nil_wf,
cons_wf,
geo-colinear-append,
use-plane-sep,
geo-left-out-1,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
length_of_nil_lemma,
istype-void,
length_of_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
lsep-iff-all-sep,
geo-sep-sym,
left-implies-sep,
geo-between-sep,
geo-between-out,
out-preserves-angle-cong_1,
geo-congruent_wf,
left-convex,
geo-sep_wf,
geo-between_wf,
left-convex2,
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-left_wf,
geo-cong-angle_wf
Rules used in proof :
inrFormation_alt,
imageElimination,
promote_hyp,
functionIsType,
equalitySymmetry,
equalityTransitivity,
int_eqEquality,
equalityIstype,
baseClosed,
imageMemberEquality,
rename,
setElimination,
lambdaEquality_alt,
approximateComputation,
unionElimination,
natural_numberEquality,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
inlFormation_alt,
independent_functionElimination,
dependent_functionElimination,
dependent_pairFormation_alt,
inhabitedIsType,
because_Cache,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
universeIsType,
productIsType,
sqequalRule,
independent_pairFormation,
hypothesis,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}b,p,b',p',a,c,a',c':Point.
((a leftof bp \mwedge{} c leftof pb)
{}\mRightarrow{} (a' leftof b'p' \mwedge{} c' leftof p'b')
{}\mRightarrow{} (abp \mcong{}\msuba{} a'b'p' \mwedge{} pbc \mcong{}\msuba{} p'b'c')
{}\mRightarrow{} abc \mcong{}\msuba{} a'b'c')
Date html generated:
2019_10_29-AM-09_19_11
Last ObjectModification:
2019_10_18-PM-03_15_03
Theory : euclidean!plane!geometry
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