Nuprl Lemma : interior-point-cong-angle-transfer-full
∀g:EuclideanPlane. ∀a,b,c,d,e,f,x,y,z:Point.
(abc < xyz
⇒ def ≅a xyz
⇒ d # ef
⇒ (∃p,p',d',f':Point. (d'ep ≅a abc ∧ d'_p'_f' ∧ p' ≠ f' ∧ (out(e dd') ∧ out(e ff')) ∧ e_p'_p ∧ (¬d_e_p))))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-lt-angle: abc < xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
geo-cong-angle: abc ≅a xyz,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
cand: A c∧ B,
not: ¬A,
false: False,
squash: ↓T,
true: True,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
geo-out: out(p ab),
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
oriented-plane: OrientedPlane
Lemmas referenced :
cong-angle-out-exists-cong3,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-angle_wf,
geo-lt-angle_wf,
geo-point_wf,
geo-out_weakening,
geo-eq_weakening,
geo-sep-sym,
out-preserves-angle-cong_1,
geo-congruent-between-exists,
geo-congruent-iff-length,
geo-between-symmetry,
euclidean-plane-axioms,
geo-congruent-symmetry,
geo-congruent-sep,
geo-out_inversion,
geo-between-trivial,
geo-between_wf,
geo-sep_wf,
geo-out_wf,
istype-void,
geo-inner-five-segment,
geo-add-length-between,
geo-length-flip,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
colinear-lsep,
out-preserves-lsep,
lsep-symmetry,
lsep-all-sym,
geo-colinear-permute,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
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-implies-sep,
geo-between-sep,
cong-tri-implies-cong-angle2,
out-cong-angle,
geo-between-out,
geo-cong-angle-symm2,
geo-cong-angle-transitivity,
stable__false,
false_wf,
not_wf,
minimal-double-negation-hyp-elim,
geo-between_functionality,
geo-sep_functionality,
geo-cong-angle_functionality,
geo-congruent_functionality,
minimal-not-not-excluded-middle,
colinear-lsep-cycle,
geo-colinear-append,
cons_wf,
nil_wf,
length_wf,
select_wf,
nat_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
l_member_wf,
geo-out-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
lsep-not-between,
geo-not-bet-and-out
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
universeIsType,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
equalitySymmetry,
independent_pairFormation,
productIsType,
functionIsType,
dependent_pairFormation_alt,
equalityTransitivity,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
unionEquality,
functionEquality,
unionIsType,
setElimination,
rename,
equalityIstype,
int_eqEquality
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f,x,y,z:Point.
(abc < xyz
{}\mRightarrow{} def \mcong{}\msuba{} xyz
{}\mRightarrow{} d \# ef
{}\mRightarrow{} (\mexists{}p,p',d',f':Point
(d'ep \mcong{}\msuba{} abc \mwedge{} d'\_p'\_f' \mwedge{} p' \mneq{} f' \mwedge{} (out(e dd') \mwedge{} out(e ff')) \mwedge{} e\_p'\_p \mwedge{} (\mneg{}d\_e\_p))))
Date html generated:
2019_10_16-PM-01_51_23
Last ObjectModification:
2019_09_27-PM-04_49_26
Theory : euclidean!plane!geometry
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