Nuprl Lemma : congruence-preserves-between_symmetric-points
∀e:BasicGeometry. ∀a,b,c,b',c':Point.
(a_b_c ⇒ Colinear(a;c;c') ⇒ ab ≅ ab' ⇒ ac ≅ ac' ⇒ b'-a-b ⇒ c ≠ c' ⇒ a_b'_c')
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
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,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
prop: ℙ,
and: P ∧ Q,
uiff: uiff(P;Q),
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
squash: ↓T,
true: True,
geo-eq: a ≡ b,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-strict-between: a-b-c
Lemmas referenced :
geo-extend-exists,
geo-sep-sym,
geo-strict-between-sep2,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
geo-sep_wf,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between_wf,
geo-congruent_wf,
geo-colinear_wf,
geo-between_wf,
geo-point_wf,
geo-congruent-iff-length,
geo-construction-unicity,
subtype_rel_self,
geo-between-sep,
geo-strict-between-sep3,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-between-exchange4,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
geo-three-segment,
geo-colinear-cases,
basic-geometry-_wf,
stable__geo-between,
geo-eq_wf,
istype-void,
geo-between-trivial2,
geo-between_functionality,
geo-eq_weakening,
geo-between-trivial,
geo-between-congruent
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesisEquality,
independent_functionElimination,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
productElimination,
rename,
universeIsType,
inhabitedIsType,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
voidElimination,
functionIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,b',c':Point.
(a\_b\_c {}\mRightarrow{} Colinear(a;c;c') {}\mRightarrow{} ab \mcong{} ab' {}\mRightarrow{} ac \mcong{} ac' {}\mRightarrow{} b'-a-b {}\mRightarrow{} c \mneq{} c' {}\mRightarrow{} a\_b'\_c')
Date html generated:
2019_10_16-PM-01_17_42
Last ObjectModification:
2019_02_16-AM-06_51_44
Theory : euclidean!plane!geometry
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