Nuprl Lemma : eu-cong3-to-conga
∀e:EuclideanPlane. ∀a,b,c,d,E,f:Point.
((∃a',c',d',f':Point. (out(b a'a) ∧ out(b c'c) ∧ out(E d'd) ∧ out(E f'f) ∧ Cong3(a'bc',d'Ef'))) ⇒ abc = dEf)
Proof
Definitions occuring in Statement :
eu-out: out(p ab),
eu-cong-tri: Cong3(abc,a'b'c'),
eu-cong-angle: abc = xyz,
euclidean-plane: EuclideanPlane,
eu-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
eu-cong-angle: abc = xyz,
not: ¬A,
eu-out: out(p ab),
cand: A c∧ B,
false: False,
eu-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
uimplies: b supposing a,
eu-five-seg-compressed: FSC(a;b;c;d a';b';c';d'),
stable: Stable{P},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
eu-colinear-set: eu-colinear-set(e;L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
exists_wf,
eu-point_wf,
eu-out_wf,
eu-cong-tri_wf,
euclidean-plane_wf,
equal_wf,
eu-extend-exists,
not_wf,
eu-between-eq_wf,
eu-congruent_wf,
eu-out-refl,
eu-cong3-to-conga-aux,
eu-congruent-iff-length,
eu-length-flip,
eu-fsc-ap,
stable__colinear,
eu-colinear_wf,
eu-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
eu-colinear-is-colinear-set,
eu-between-eq-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
eu-between-eq-exchange3,
eu-between-eq-symmetry,
eu-between-eq-exchange4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
hypothesisEquality,
independent_pairFormation,
independent_functionElimination,
equalitySymmetry,
voidElimination,
dependent_functionElimination,
dependent_set_memberEquality,
dependent_pairFormation,
independent_isectElimination,
equalityTransitivity,
inlFormation,
inrFormation,
isect_memberEquality,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,E,f:Point.
((\mexists{}a',c',d',f':Point. (out(b a'a) \mwedge{} out(b c'c) \mwedge{} out(E d'd) \mwedge{} out(E f'f) \mwedge{} Cong3(a'bc',d'Ef')))
{}\mRightarrow{} abc = dEf)
Date html generated:
2016_10_26-AM-07_45_33
Last ObjectModification:
2016_08_04-PM-06_59_09
Theory : euclidean!geometry
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