Nuprl Lemma : geo-lt-add1_2
∀e:BasicGeometry. ∀p,q,r:{a:Point| O_X_a} . (X ≠ p ⇒ X ≠ q ⇒ X ≠ r ⇒ p < q ⇒ p + r < q + r)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
basic-geometry: BasicGeometry,
geo-X: X,
geo-O: O,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-lt: p < q,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
geo-le: p ≤ q,
squash: ↓T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
prop: ℙ,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
basic-geometry-: BasicGeometry-,
geo-length-type: Length,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
respects-equality: respects-equality(S;T),
geo-strict-between: a-b-c,
uiff: uiff(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,
geo-eq: a ≡ b
Lemmas referenced :
sq_stable__geo-le,
geo-add-length_wf,
subtype-geo-length-type,
geo-length_wf,
geo-mk-seg_wf,
geo-sep_wf,
geo-le_wf,
geo-lt_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-X_wf,
geo-point_wf,
geo-between_wf,
geo-O_wf,
geo-le-iff-between-points,
squash_wf,
true_wf,
geo-length-type_wf,
subtype_rel_self,
iff_weakening_equal,
geo-le-add1,
geo-between-sep,
geo-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep3,
geo-proper-extend-exists,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-exchange4,
geo-between-outer-trans,
respects-equality-quotient1,
geo-eq_wf,
geo-length-equiv,
respects-equality-set-trivial,
geo-add-length-between,
geo-length-equality,
geo-congruent-iff-length,
equal_wf,
istype-universe,
geo-sep-sym,
geo-add-length-comm,
geo-add-length-assoc,
oriented-colinear-append,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-colinear-is-colinear-set,
geo-strict-between-implies-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,
geo-colinear-cases,
basic-geometry-_wf,
stable__geo-between,
geo-strict-between_wf,
geo-between-trivial2,
geo-between_functionality,
geo-eq_weakening,
geo-between-trivial,
geo-strict-between-sep1,
geo-strict-between_functionality,
geo-between-exchange3,
geo-add-length-cancel-left-le,
geo-le_antisymmetry,
geo-add-length-cancel-left,
geo-construction-unicity-from-first,
geo-between-implies-out2,
geo-not-bet-and-out
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
hypothesisEquality,
cut,
hypothesis,
independent_pairFormation,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
independent_functionElimination,
imageMemberEquality,
baseClosed,
productIsType,
universeIsType,
instantiate,
independent_isectElimination,
dependent_functionElimination,
inhabitedIsType,
setIsType,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
universeEquality,
dependent_set_memberEquality_alt,
equalityIstype,
setEquality,
inlFormation_alt,
inrFormation_alt,
isect_memberEquality_alt,
voidElimination,
unionElimination,
approximateComputation,
functionIsType,
applyLambdaEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,q,r:\{a:Point| O\_X\_a\} . (X \mneq{} p {}\mRightarrow{} X \mneq{} q {}\mRightarrow{} X \mneq{} r {}\mRightarrow{} p < q {}\mRightarrow{} p + r < q + r)
Date html generated:
2019_10_16-PM-01_35_36
Last ObjectModification:
2019_01_11-PM-03_18_43
Theory : euclidean!plane!geometry
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