Nuprl Lemma : geo-Aax4
∀g:EuclideanPlane. ∀a,b,c:Point. ∀l,m:Line. (a ≠ b ⇒ (a I l ∧ b I l) ⇒ ((a I m ∧ c I m) ∧ c # l) ⇒ b # m)
Proof
Definitions occuring in Statement :
geo-plsep: p # l,
geo-incident: p I L,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
geo-incident: p I L,
cand: A c∧ B,
oriented-plane: OrientedPlane,
or: P ∨ Q,
euclidean-plane: EuclideanPlane,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
pi2: snd(t),
pi1: fst(t),
geo-plsep: p # l,
geo-line: Line,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-colinear_wf,
top_wf,
subtype_rel_product,
pi1_wf_top,
and_wf,
colinear-lsep,
geo-sep-sym,
list_ind_nil_lemma,
list_ind_cons_lemma,
exists_wf,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
oriented-colinear-append,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geoline_wf,
trivial-equal,
lsep-all-sym,
colinear-lsep',
geo-sep-or,
geo-point_wf,
geo-line_wf,
geo-sep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-plsep_wf,
geoline-subtype1,
geo-incident_wf
Rules used in proof :
imageElimination,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
independent_pairEquality,
lambdaEquality,
inlFormation,
inrFormation,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_pairEquality,
independent_functionElimination,
unionElimination,
dependent_set_memberEquality,
rename,
setElimination,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
productEquality,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c:Point. \mforall{}l,m:Line.
(a \mneq{} b {}\mRightarrow{} (a I l \mwedge{} b I l) {}\mRightarrow{} ((a I m \mwedge{} c I m) \mwedge{} c \# l) {}\mRightarrow{} b \# m)
Date html generated:
2018_05_23-PM-06_09_22
Last ObjectModification:
2018_05_23-AM-10_44_53
Theory : euclidean!plane!geometry
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