Nuprl Lemma : geo-congruent-preserves-gt
∀e:BasicGeometry. ∀a,b,c,d,a',b',c',d':Point. (ab > cd ⇒ ab ≅ a'b' ⇒ cd ≅ c'd' ⇒ a'b' > c'd')
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-gt: cd > ab,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-gt: cd > ab,
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
cand: A c∧ B,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
geo-congruent-between-exists,
geo-sep-sym,
geo-congruent-iff-length,
geo-length-flip,
geo-between-symmetry,
geo-congruent-symmetry,
geo-congruent-sep,
geo-between_wf,
geo-congruent_wf,
geo-sep_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-gt_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
cut,
productElimination,
thin,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
independent_isectElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
independent_pairFormation,
sqequalRule,
productIsType,
universeIsType,
applyEquality,
imageMemberEquality,
baseClosed,
instantiate,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,a',b',c',d':Point. (ab > cd {}\mRightarrow{} ab \mcong{} a'b' {}\mRightarrow{} cd \mcong{} c'd' {}\mRightarrow{} a'b' > c'd')
Date html generated:
2019_10_16-PM-01_16_59
Last ObjectModification:
2019_02_15-AM-06_11_47
Theory : euclidean!plane!geometry
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