Nuprl Lemma : geo-SS_functionality
∀[g:EuclideanPlane]. ∀[a,b:Point]. ∀[u:{u:Point| u leftof ab} ]. ∀[v:{v:Point| v leftof ba} ]. ∀[a',b':Point].
∀[u':{u:Point| u leftof a'b'} ]. ∀[v':{v:Point| v leftof b'a'} ].
(geo-SS(g;a;b;u;v) ≡ geo-SS(g;a';b';u';v')) supposing (v ≡ v' and u ≡ u' and b ≡ b' and a ≡ a')
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-SS: geo-SS(g;a;b;u;v),
geo-eq: a ≡ b,
geo-left: a leftof bc,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
subtract: n - m,
cons: [a / b],
select: L[n],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
rev_implies: P ⇐ Q,
cand: A c∧ B,
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
oriented-plane: OrientedPlane,
prop: ℙ,
false: False,
not: ¬A,
geo-eq: a ≡ b,
squash: ↓T,
sq_stable: SqStable(P),
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
euclidean-plane: EuclideanPlane,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
left-right-sep,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
length_of_nil_lemma,
length_of_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
geo-colinear_wf,
not-lsep-iff-colinear,
istype-void,
geo-lsep_wf,
lsep-all-sym2,
geo-intersection-unicity,
geo-between_functionality,
geo-left_functionality,
geo-eq_weakening,
geo-colinear_functionality,
geo-point_wf,
geo-left_wf,
geo-eq_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
sq_stable__geo-eq,
geo-SS_wf
Rules used in proof :
productIsType,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
natural_numberEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
functionIsType,
promote_hyp,
voidElimination,
setIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
universeIsType,
functionIsTypeImplies,
lambdaEquality_alt,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
productElimination,
because_Cache,
lambdaFormation_alt,
inhabitedIsType,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[g:EuclideanPlane]. \mforall{}[a,b:Point]. \mforall{}[u:\{u:Point| u leftof ab\} ]. \mforall{}[v:\{v:Point| v leftof ba\} ].
\mforall{}[a',b':Point]. \mforall{}[u':\{u:Point| u leftof a'b'\} ]. \mforall{}[v':\{v:Point| v leftof b'a'\} ].
(geo-SS(g;a;b;u;v) \mequiv{} geo-SS(g;a';b';u';v')) supposing (v \mequiv{} v' and u \mequiv{} u' and b \mequiv{} b' and a \mequiv{} a')
Date html generated:
2019_10_29-AM-09_16_34
Last ObjectModification:
2019_10_18-PM-03_36_51
Theory : euclidean!plane!geometry
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