Nuprl Lemma : geo-CC_functionality
∀[g:EuclideanPlane]. ∀[a,b:Point]. ∀[c:{c:Point| a ≠ c} ]. ∀[d:{d:Point| StrictOverlap(a;b;c;d)} ]. ∀[a',b':Point].
∀[c':{c':Point| a' ≠ c'} ]. ∀[d':{d':Point| StrictOverlap(a';b';c';d')} ].
(CC(a;b;c;d) ≡ CC(a';b';c';d')) supposing (d ≡ d' and c ≡ c' and b ≡ b' and a ≡ a')
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-CC: CC(a;b;c;d),
circle-strict-overlap: StrictOverlap(a;b;c;d),
geo-eq: a ≡ b,
geo-sep: a ≠ b,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-eq: a ≡ b,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
oriented-plane: OrientedPlane,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q
Lemmas referenced :
geo-CC_wf,
geo-eq_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
circle-strict-overlap_wf,
geo-sep_wf,
geo-point_wf,
sq_stable__geo-eq,
geo-eq_inversion,
Euclid-Prop7,
geo-left_wf,
geo-congruent-iff-length,
geo-left_functionality,
geo-eq_weakening,
geo-congruent_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
because_Cache,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality_alt,
functionIsTypeImplies,
universeIsType,
applyEquality,
instantiate,
independent_isectElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
setIsType,
voidElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt
Latex:
\mforall{}[g:EuclideanPlane]. \mforall{}[a,b:Point]. \mforall{}[c:\{c:Point| a \mneq{} c\} ]. \mforall{}[d:\{d:Point| StrictOverlap(a;b;c;d)\} ].
\mforall{}[a',b':Point]. \mforall{}[c':\{c':Point| a' \mneq{} c'\} ]. \mforall{}[d':\{d':Point| StrictOverlap(a';b';c';d')\} ].
(CC(a;b;c;d) \mequiv{} CC(a';b';c';d')) supposing (d \mequiv{} d' and c \mequiv{} c' and b \mequiv{} b' and a \mequiv{} a')
Date html generated:
2019_10_16-PM-01_48_22
Last ObjectModification:
2019_07_16-AM-11_38_02
Theory : euclidean!plane!geometry
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