Nuprl Lemma : interior-point-preserves-cong-angle
∀g:EuclideanPlane. ∀a,b,c,x,y,z,p,q:Point.
(abc ≅a xyz ⇒ a_q_c ⇒ x_p_z ⇒ Cong3(abc,xyz) ⇒ Cong3(aqc,xpz) ⇒ x # yz ⇒ abq ≅a xyp)
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-between: a_b_c,
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,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
geo-cong-tri: Cong3(abc,a'b'c'),
and: P ∧ Q,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
geo-cong-angle: abc ≅a xyz,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
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,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-inner-five-segment,
geo-between-symmetry,
geo-congruent-iff-length,
geo-length-flip,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-tri_wf,
geo-between_wf,
geo-cong-angle_wf,
geo-point_wf,
geo-sep-or,
lsep-implies-sep,
geo-sep_wf,
colinear-lsep,
lsep-symmetry,
lsep-all-sym,
euclidean-plane-axioms,
geo-colinear-permute,
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,
geo-congruent-sep,
lsep-symmetry2,
geo-between-trivial,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_isectElimination,
because_Cache,
hypothesis,
productElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType,
setElimination,
rename,
independent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productIsType,
dependent_pairFormation_alt
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z,p,q:Point.
(abc \mcong{}\msuba{} xyz {}\mRightarrow{} a\_q\_c {}\mRightarrow{} x\_p\_z {}\mRightarrow{} Cong3(abc,xyz) {}\mRightarrow{} Cong3(aqc,xpz) {}\mRightarrow{} x \# yz {}\mRightarrow{} abq \mcong{}\msuba{} xyp)
Date html generated:
2019_10_16-PM-01_50_51
Last ObjectModification:
2018_11_20-AM-10_52_51
Theory : euclidean!plane!geometry
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