Nuprl Lemma : eu-add-length-comm
∀[e:EuclideanPlane]. ∀[x,y:{p:Point| O_X_p} ]. (x + y = y + x ∈ {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,
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,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
eu-add-length: p + q,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
false: False,
stable: Stable{P},
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
eu-between-eq_wf,
eu-O_wf,
eu-X_wf,
set_wf,
eu-point_wf,
eu-not-colinear-OXY,
eu-between-eq-same,
equal_wf,
eu-extend-property,
not_wf,
eu-extend_wf,
eu-congruent_wf,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-between-eq-exchange4,
eu-construction-unicity,
stable__eu-congruent,
eu-congruence-identity,
eu-congruent-iff-length,
eu-length-flip,
eu-three-segment,
eu-add-length_wf,
sq_stable__eu-between-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
isect_memberEquality,
axiomEquality,
productElimination,
lambdaFormation,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
equalityTransitivity,
independent_isectElimination,
independent_functionElimination,
voidElimination,
productEquality,
equalityEquality,
promote_hyp,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[x,y:\{p:Point| O\_X\_p\} ]. (x + y = y + x)
Date html generated:
2016_10_26-AM-07_41_54
Last ObjectModification:
2016_07_12-AM-08_08_14
Theory : euclidean!geometry
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