Nuprl Lemma : geo-cong3-to-conga
∀e:BasicGeometry. ∀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 ≅a dEf)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-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,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-cong-angle: abc ≅a xyz,
cand: A c∧ B,
basic-geometry: BasicGeometry,
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
squash: ↓T,
true: True,
geo-five-seg-compressed: FSC(a;b;c;d a';b';c';d'),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
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
Lemmas referenced :
geo-out_wf,
geo-cong-tri_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep-sym,
geo-extend-exists,
geo-between_wf,
geo-congruent_wf,
geo-out_inversion,
geo-cong3-to-conga-aux,
geo-congruent-iff-length,
geo-length-flip,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
geo-add-length-comm,
geo-fsc-ap,
oriented-colinear-append,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-sep_wf,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
geo-out-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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
productIsType,
because_Cache,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
rename,
dependent_pairFormation_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inlFormation_alt,
inrFormation_alt,
equalityIstype,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation
Latex:
\mforall{}e:BasicGeometry. \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 \mcong{}\msuba{} dEf)
Date html generated:
2019_10_16-PM-01_28_45
Last ObjectModification:
2018_12_12-PM-02_31_47
Theory : euclidean!plane!geometry
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