Nuprl Lemma : out-preserves-angle-cong_1
∀e:BasicGeometry. ∀a,b,c,a',b',c',p,q,p',q':Point.
(abc ≅a a'b'c' ⇒ out(b cq) ⇒ out(b ap) ⇒ out(b' c'q') ⇒ out(b' a'p') ⇒ pbq ≅a p'b'q')
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
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,
basic-geometry: BasicGeometry,
geo-out: out(p ab),
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-cong-tri: Cong3(abc,a'b'c'),
cand: A c∧ B,
uiff: uiff(P;Q)
Lemmas referenced :
cong-angle-out-exists3,
geo-sep-sym,
geo-out_wf,
geo-cong-angle_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent-iff-length,
geo-length-flip,
geo-out_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
productElimination,
universeIsType,
isectElimination,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',b',c',p,q,p',q':Point.
(abc \mcong{}\msuba{} a'b'c' {}\mRightarrow{} out(b cq) {}\mRightarrow{} out(b ap) {}\mRightarrow{} out(b' c'q') {}\mRightarrow{} out(b' a'p') {}\mRightarrow{} pbq \mcong{}\msuba{} p'b'q')
Date html generated:
2019_10_16-PM-01_26_40
Last ObjectModification:
2018_10_09-AM-10_13_14
Theory : euclidean!plane!geometry
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