Nuprl Lemma : geo-intersect-iff
∀e:EuclideanPlane. ∀P,L:LINE.
(P \/ L ⇐⇒ ∃a,b,c,d,v:Point. (a-v-b ∧ c-v-d ∧ a I P ∧ b I P ∧ c I L ∧ d I L ∧ a leftof cd ∧ b leftof dc))
Proof
Definitions occuring in Statement :
geo-intersect: L \/ M,
geo-incident: p I L,
geoline: LINE,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-intersect: L \/ M,
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-line: Line,
pi2: snd(t),
pi1: fst(t),
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
cand: A c∧ B,
squash: ↓T,
geo-incident: p I L,
true: True,
or: P ∨ Q,
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
oriented-plane: OrientedPlane,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
respects-equality: respects-equality(S;T)
Lemmas referenced :
geo-intersect_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between_wf,
geo-incident_wf,
geo-left_wf,
geoline_wf,
exists_wf,
and_wf,
geoline-subtype1,
geo-SS_wf,
sq_stable__and,
geo-colinear_wf,
geo-between_wf,
sq_stable__colinear,
sq_stable__geo-between,
geo-sep_wf,
trivial-equal,
geo-sep-or,
symmetric-point-construction,
geo-sep-sym,
colinear-lsep-cycle,
lsep-all-sym2,
geo-between-sep,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
list_ind_cons_lemma,
istype-void,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
lsep-all-sym,
geo-colinear-same,
geo-congruent-symmetry,
geo-congruent-sep,
left-implies-sep,
left-between,
not-lsep-iff-colinear,
geo-between-symmetry,
iff_weakening_uiff,
pi1_wf_top,
pi2_wf,
subtype_rel_product,
top_wf,
geo-incident-line,
geo-strict-between-implies-colinear,
lsep-colinear-sep,
geo-lsep_wf,
geo-strict-between-sep1,
geo-line-eq-geoline,
subtype-respects-equality,
geo-line_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
productIsType,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
productElimination,
promote_hyp,
equalitySymmetry,
rename,
hyp_replacement,
applyLambdaEquality,
lambdaEquality_alt,
setElimination,
dependent_set_memberEquality_alt,
equalityIstype,
equalityTransitivity,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality_alt,
productEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairEquality_alt,
independent_pairEquality,
natural_numberEquality,
unionElimination,
dependent_pairFormation_alt,
inlFormation_alt,
inrFormation_alt,
voidElimination,
approximateComputation,
functionIsType,
setIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}P,L:LINE.
(P \mbackslash{}/ L
\mLeftarrow{}{}\mRightarrow{} \mexists{}a,b,c,d,v:Point. (a-v-b \mwedge{} c-v-d \mwedge{} a I P \mwedge{} b I P \mwedge{} c I L \mwedge{} d I L \mwedge{} a leftof cd \mwedge{} b leftof dc))
Date html generated:
2019_10_16-PM-02_40_37
Last ObjectModification:
2018_12_11-PM-11_05_21
Theory : euclidean!plane!geometry
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