Nuprl Lemma : cong3-in-half-plane
∀e:EuclideanPlane. ∀a,b,c,x,y,u:Point.
(c # ab ⇒ u # xy ⇒ ab ≅ xy ⇒ (∃z:Point. (Cong3(abc,xyz) ∧ z # xy ∧ (u leftof xy ⇐⇒ z leftof xy))))
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
geo-lsep: a # bc,
geo-eq: a ≡ b,
stable: Stable{P},
uiff: uiff(P;Q),
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-cong-tri: Cong3(abc,a'b'c'),
cand: A c∧ B,
oriented-plane: OrientedPlane,
squash: ↓T,
sq_stable: SqStable(P),
basic-geometry: BasicGeometry,
sq_exists: ∃x:A [B[x]],
subtract: n - m,
cons: [a / b],
select: L[n],
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
iff: P ⇐⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
guard: {T},
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
not-left-and-right,
geo-cong-tri_wf,
lsep-all-sym,
colinear-lsep-cycle,
geo-sep-or,
geo-between-symmetry,
lsep-opposite-iff,
geo-between_wf,
geo-between_functionality,
geo-length-flip,
geo-left_functionality,
geo-congruence-identity,
minimal-not-not-excluded-middle,
geo-lsep_functionality,
geo-colinear_functionality,
geo-perp-in_functionality,
geo-eq_weakening,
geo-congruent_functionality,
minimal-double-negation-hyp-elim,
geo-congruent-iff-length,
geo-congruent-sep,
geo-congruent-symmetry,
geo-between-implies-colinear,
geo-colinear-same,
right-angle-SAS,
not_wf,
false_wf,
stable__geo-congruent,
geo-sep-sym,
geo-extend-exists,
lsep-all-sym2,
left-right-sep,
sq_stable__geo-left,
sq_stable__geo-perp-in,
sq_stable__and,
geo-left_wf,
geo-perp-in_wf,
Euclid-erect-2perp,
colinear-cong3,
geo-point_wf,
geo-lsep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-congruent_wf,
geo-colinear_wf,
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,
istype-void,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
lsep-iff-all-sep,
geo-sep_wf,
lsep-implies-sep,
Euclid-drop-perp-1
Rules used in proof :
equalityTransitivity,
promote_hyp,
unionIsType,
equalitySymmetry,
functionEquality,
unionEquality,
imageElimination,
baseClosed,
imageMemberEquality,
productEquality,
setElimination,
rename,
instantiate,
functionIsType,
inhabitedIsType,
productIsType,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
independent_pairFormation,
natural_numberEquality,
voidElimination,
isect_memberEquality_alt,
sqequalRule,
applyEquality,
isectElimination,
universeIsType,
dependent_set_memberEquality_alt,
productElimination,
hypothesis,
independent_functionElimination,
because_Cache,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,u:Point.
(c \# ab
{}\mRightarrow{} u \# xy
{}\mRightarrow{} ab \mcong{} xy
{}\mRightarrow{} (\mexists{}z:Point. (Cong3(abc,xyz) \mwedge{} z \# xy \mwedge{} (u leftof xy \mLeftarrow{}{}\mRightarrow{} z leftof xy))))
Date html generated:
2019_10_29-AM-09_18_22
Last ObjectModification:
2019_10_18-PM-03_15_17
Theory : euclidean!plane!geometry
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