Nuprl Lemma : Euclid-Prop6
∀e:EuclideanPlane. ∀a,b,c:Point. (c # ab ⇒ cab ≅a cba ⇒ ca ≅ cb)
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
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,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
oriented-plane: OrientedPlane,
iff: P ⇐⇒ Q,
euclidean-plane: EuclideanPlane,
not: ¬A,
false: False,
or: P ∨ Q,
stable: Stable{P},
rev_implies: 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,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-lsep: a # bc,
geo-eq: a ≡ b
Lemmas referenced :
Euclid-Prop6-lemma,
lsep-all-sym2,
geo-left_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-angle_wf,
geo-point_wf,
left-implies-sep,
geo-sep-sym,
cong-tri-implies-cong-angle,
geo-congruent-iff-length,
geo-length-flip,
geo-cong-angle-transitivity,
Euclid-Prop7,
geo-eq_inversion,
geo-left_functionality,
geo-eq_weakening,
geo-congruent_functionality,
geo-colinear_functionality,
stable__geo-congruent,
false_wf,
or_wf,
geo-sep_wf,
not_wf,
geo-congruent_wf,
istype-void,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
not-lsep-iff-colinear,
euclidean-plane-axioms,
geo-cong-angle-symm2,
geo-lsep_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
productElimination,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
rename,
dependent_set_memberEquality_alt,
setElimination,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
voidElimination,
dependent_pairFormation_alt,
inrFormation_alt,
inlFormation_alt,
equalityIsType1,
productIsType,
isect_memberEquality_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (c \# ab {}\mRightarrow{} cab \mcong{}\msuba{} cba {}\mRightarrow{} ca \mcong{} cb)
Date html generated:
2019_10_16-PM-01_52_07
Last ObjectModification:
2018_11_07-PM-01_01_28
Theory : euclidean!plane!geometry
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