Nuprl Lemma : geo-right-angles-congruent
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point. (((Rabc ∧ Rxyz) ∧ (x ≠ y ∧ y ≠ z) ∧ a ≠ b ∧ b ≠ c) ⇒ abc ≅a xyz)
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
right-angle: Rabc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
subtract: n - m,
cons: [a / b],
select: L[n],
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
geo-proper-extend-exists,
geo-sep-sym,
right-angle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf,
geo-add-length-between,
geo-congruent-iff-length,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
geo-length-flip,
right-angle-SAS,
geo-strict-between-sep1,
geo-colinear-permute,
euclidean-plane-axioms,
right-angle-symmetry,
less_than_wf,
le_wf,
istype-false,
length_of_nil_lemma,
istype-void,
length_of_cons_lemma,
geo-strict-between-implies-colinear,
geo-colinear-is-colinear-set,
right-angle-colinear,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-between_wf,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
rename,
productIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
dependent_pairFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(((Rabc \mwedge{} Rxyz) \mwedge{} (x \mneq{} y \mwedge{} y \mneq{} z) \mwedge{} a \mneq{} b \mwedge{} b \mneq{} c) {}\mRightarrow{} abc \mcong{}\msuba{} xyz)
Date html generated:
2019_10_16-PM-01_52_46
Last ObjectModification:
2018_12_15-PM-10_04_57
Theory : euclidean!plane!geometry
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