Nuprl Lemma : congruence-preserves-right-angle
∀e:BasicGeometry. ∀a,b,c:Point. (Rabc ⇒ (∀a',b',c':Point. (Cong3(abc,a'b'c') ⇒ Ra'b'c')))
Proof
Definitions occuring in Statement :
right-angle: Rabc,
geo-cong-tri: Cong3(abc,a'b'c'),
basic-geometry: BasicGeometry,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
false: False,
or: P ∨ Q,
not: ¬A,
stable: Stable{P},
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
geo-cong-tri: Cong3(abc,a'b'c'),
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
right-angle: Rabc,
uiff: uiff(P;Q),
cand: A c∧ B,
geo-midpoint: a=m=b
Lemmas referenced :
minimal-not-not-excluded-middle,
right-angle_functionality,
geo-eq_weakening,
geo-congruent_functionality,
minimal-double-negation-hyp-elim,
stable__right-angle,
geo-congruent_wf,
geo-sep_wf,
or_wf,
false_wf,
not_wf,
stable__not,
right-angle_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
geo-cong-tri_wf,
geo-congruent-sep,
geo-congruent-symmetry,
symmetric-point-construction,
geo-midpoint-symmetry,
implies-right-angle,
geo-length-flip,
geo-congruent-iff-length,
geo-sep-sym,
geo-five-segment,
geo-congruence-identity,
right-angle-trivial
Rules used in proof :
andLevelFunctionality,
promote_hyp,
levelHypothesis,
independent_pairFormation,
addLevel,
voidElimination,
unionElimination,
independent_functionElimination,
dependent_functionElimination,
functionEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
rename,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (Rabc {}\mRightarrow{} (\mforall{}a',b',c':Point. (Cong3(abc,a'b'c') {}\mRightarrow{} Ra'b'c')))
Date html generated:
2017_10_02-PM-06_42_19
Last ObjectModification:
2017_08_05-PM-04_48_29
Theory : euclidean!plane!geometry
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