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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