Nuprl Lemma : cong-angle-out-aux-weak
∀g:HeytingGeometry. ∀a,b,c,d,e,f,a',c',d',f':Point.
(abc ≅a def ⇒ out(b a'a) ⇒ out(b c'c) ⇒ out(e d'd) ⇒ out(e f'f) ⇒ ba' ≅ ed' ⇒ bc' ≅ ef' ⇒ a'c' ≅ d'f')
Proof
Definitions occuring in Statement :
heyting-geometry: HeytingGeometry,
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
heyting-geometry: HeytingGeometry,
prop: ℙ,
geo-out: out(p ab),
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
uiff: uiff(P;Q)
Lemmas referenced :
geo-between-out,
euclidean-plane-subtype-basic,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
basic-geometry_wf,
geo-sep-sym,
geo-between-sep,
geo-out_transitivity,
geo-out_inversion,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-out_wf,
geo-cong-angle_wf,
geo-point_wf,
geo-out-cong-cong,
geo-colinear-five-segment,
geo-colinear-is-colinear-set,
geo-out-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
oriented-colinear-append,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-sep_wf,
geo-between-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-congruent-iff-length,
geo-length-flip
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
because_Cache,
universeIsType,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
dependent_pairFormation_alt,
inlFormation_alt,
inrFormation_alt,
equalityIsType1,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}g:HeytingGeometry. \mforall{}a,b,c,d,e,f,a',c',d',f':Point.
(abc \mcong{}\msuba{} def
{}\mRightarrow{} out(b a'a)
{}\mRightarrow{} out(b c'c)
{}\mRightarrow{} out(e d'd)
{}\mRightarrow{} out(e f'f)
{}\mRightarrow{} ba' \mcong{} ed'
{}\mRightarrow{} bc' \mcong{} ef'
{}\mRightarrow{} a'c' \mcong{} d'f')
Date html generated:
2019_10_16-PM-02_08_01
Last ObjectModification:
2018_11_07-PM-01_09_08
Theory : euclidean!plane!geometry
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