Nuprl Lemma : Euclid-Prop9
∀e:EuclideanPlane. ∀a,b:Point. ∀c:{c:Point| c # ba} . ∃f:Point. acf ≅a bcf
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
true: True,
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,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
select: L[n],
cons: [a / b],
subtract: n - m,
sq_exists: ∃x:A [B[x]],
oriented-plane: OrientedPlane,
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
sq_stable__geo-lsep,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
lsep-implies-sep,
geo-extend-exists,
geo-congruent-iff-length,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
colinear-lsep-cycle,
lsep-all-sym,
geo-sep-sym,
geo-between-sep,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
Euclid-Prop10,
geo-sep_wf,
geo-cong-angle_wf,
sq_stable__geo-sep,
colinear-lsep',
geo-strict-between-sep2,
geo-strict-between-implies-colinear,
geo-between-trivial,
geo-congruent-refl,
geo-length-flip,
geo-between_wf,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
setIsType,
inhabitedIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
productElimination,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
dependent_pairFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. \mforall{}c:\{c:Point| c \# ba\} . \mexists{}f:Point. acf \mcong{}\msuba{} bcf
Date html generated:
2019_10_16-PM-01_41_36
Last ObjectModification:
2018_11_07-PM-01_00_09
Theory : euclidean!plane!geometry
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