Nuprl Lemma : eu-add-length-cancel-left
∀[e:EuclideanPlane]. ∀[x,y,z:{p:Point| O_X_p} ]. x = y ∈ {p:Point| O_X_p} supposing z + x = z + y ∈ {p:Point| O_X_p}
Proof
Definitions occuring in Statement :
eu-add-length: p + q,
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-X: X,
eu-O: O,
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
eu-add-length: p + q,
euclidean-plane: EuclideanPlane,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
false: False,
prop: ℙ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q)
Lemmas referenced :
eu-not-colinear-OXY,
eu-between-eq-same2,
eu-X_wf,
equal_wf,
eu-point_wf,
eu-O_wf,
eu-extend-property,
not_wf,
eu-extend_wf,
and_wf,
eu-between-eq_wf,
eu-congruent_wf,
eu-add-length_wf,
set_wf,
euclidean-plane_wf,
eu-construction-unicity,
eu-congruent-iff-length,
eu-mk-seg_wf,
eu-segment_wf,
eu-length_wf,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-between-eq-exchange4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
productElimination,
hypothesis,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality,
because_Cache,
equalityEquality,
setEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
lambdaEquality,
independent_pairFormation,
applyEquality
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[x,y,z:\{p:Point| O\_X\_p\} ]. x = y supposing z + x = z + y
Date html generated:
2016_05_18-AM-06_38_31
Last ObjectModification:
2015_12_28-AM-09_24_26
Theory : euclidean!geometry
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