Nuprl Lemma : geo-out-cong-implies-eq
∀e:BasicGeometry. ∀a,b,x,y:Point. (out(a bx) ⇒ out(a by) ⇒ ax ≅ ay ⇒ x ≡ y)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
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,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-out: out(p ab),
l_member: (x ∈ l),
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
false: False,
select: L[n],
cons: [a / b],
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
ge: i ≥ j ,
subtract: n - m,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
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,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
guard: {T},
geo-eq: a ≡ b,
geo-strict-between: a-b-c
Lemmas referenced :
geo-colinear-append,
cons_wf,
geo-point_wf,
nil_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
length_of_cons_lemma,
length_of_nil_lemma,
istype-less_than,
length_wf,
select_wf,
nat_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
geo-sep_wf,
l_member_wf,
geo-colinear-is-colinear-set,
geo-out-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
geo-colinear-cases,
subtype_rel_self,
basic-geometry-_wf,
geo-eq_wf,
stable_geo-eq,
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-strict-between_wf,
geo-congruent_wf,
geo-out_wf,
geo-sep-sym,
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
geo-sep-O-X,
geo-construction-unicity,
euclidean-plane-axioms,
geo-strict-between-sep3,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-congruent-symmetry,
geo-between-exchange3,
geo-not-bet-and-out,
geo-out_inversion,
geo-out_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_functionElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
imageMemberEquality,
baseClosed,
inhabitedIsType,
productIsType,
setElimination,
rename,
equalityIstype,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
functionIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,x,y:Point. (out(a bx) {}\mRightarrow{} out(a by) {}\mRightarrow{} ax \mcong{} ay {}\mRightarrow{} x \mequiv{} y)
Date html generated:
2019_10_16-PM-01_24_39
Last ObjectModification:
2019_03_14-PM-09_08_17
Theory : euclidean!plane!geometry
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