Nuprl Lemma : geo-line-eq_transitivity
∀g:EuclideanPlane. ∀l,m,n:Line. (l ≡ m ⇒ m ≡ n ⇒ l ≡ n)
Proof
Definitions occuring in Statement :
geo-line-eq: l ≡ m,
geo-line: Line,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
not: ¬A,
geo-line-eq: l ≡ m,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
guard: {T},
geo-line-sep: geo-line-sep(g;l;m),
pi2: snd(t),
pi1: fst(t),
geo-line: Line,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
oriented-plane: OrientedPlane,
geo-colinear: Colinear(a;b;c)
Lemmas referenced :
euclidean-plane_wf,
geo-line_wf,
geo-line-eq_wf,
geo-line-sep_wf,
geo-line-eq-to-col,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
geo-colinear-is-colinear-set,
exists_wf,
geo-sep_wf,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
cons_wf,
oriented-colinear-append,
geo-lsep_wf,
geo-colinear_wf,
lsep-all-sym,
lsep-not-between
Rules used in proof :
dependent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaEquality,
productEquality,
inlFormation,
inrFormation,
independent_pairFormation,
dependent_pairFormation,
independent_isectElimination,
instantiate,
applyEquality
Latex:
\mforall{}g:EuclideanPlane. \mforall{}l,m,n:Line. (l \mequiv{} m {}\mRightarrow{} m \mequiv{} n {}\mRightarrow{} l \mequiv{} n)
Date html generated:
2018_05_22-PM-01_02_07
Last ObjectModification:
2018_01_17-PM-07_58_30
Theory : euclidean!plane!geometry
Home
Index