Nuprl Lemma : eu-eq_dist-axiomA10
∀G:EuclideanPlane. ∀a,b,c,d,e,f,g:Point. (D(a;b;c;d;e;f) ⇒ D(c;d;e;f;a;b) ⇒ c ≠ d)
Proof
Definitions occuring in Statement :
dist: D(a;b;c;d;e;f),
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
basic-geometry: BasicGeometry,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
squash: ↓T,
true: True,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
false: False,
geo-length: |s|,
top: Top,
cand: A c∧ B
Lemmas referenced :
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
dist_wf,
geo-add-length-lt-sep,
dist-lemma-lt,
geo-add-length-lt-sep2,
geo-sep-or,
geo-length_wf1,
geo-mk-seg_wf,
geo-add-length_wf1,
geo-sep_wf,
geo-sep-iff-or-lt,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-zero2,
geo-length_wf,
geo-add-length_wf,
subtype_rel_self,
iff_weakening_equal,
geo-add-length-comm,
geo-add-length-cancel-left-lt,
geo-lt-sep,
geo-lt-null-segment,
geo-congruent-sep,
geo-X_wf,
geo_seg2_mk_seg_lemma,
istype-void,
geo_seg1_mk_seg_lemma,
geo-extend-property,
geo-O_wf,
geo-sep-O-X,
geo-between-same-side-or,
geo-le_wf,
geo-le-add1,
geo-le-iff-between-points,
geo-le_weakening-lt,
geo-lt_transitivity2,
geo-add-length-zero3,
geo-add-length-cancel-right-lt
Rules used in proof :
independent_isectElimination,
instantiate,
applyEquality,
inhabitedIsType,
rename,
setElimination,
isectElimination,
universeIsType,
unionElimination,
because_Cache,
sqequalRule,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
hyp_replacement,
applyLambdaEquality,
voidElimination,
isect_memberEquality_alt,
independent_pairFormation
Latex:
\mforall{}G:EuclideanPlane. \mforall{}a,b,c,d,e,f,g:Point. (D(a;b;c;d;e;f) {}\mRightarrow{} D(c;d;e;f;a;b) {}\mRightarrow{} c \mneq{} d)
Date html generated:
2019_10_16-PM-02_58_18
Last ObjectModification:
2019_02_28-PM-05_37_45
Theory : euclidean!plane!geometry
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