Nuprl Lemma : ip-five-segment
∀[rv:InnerProductSpace]. ∀[a,b,c,d,A,B,C,D:Point].
(cd=CD) supposing (bd=BD and ad=AD and bc=BC and ab=AB and A_B_C and a_b_c and a # b)
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
ip-congruent: ab=cd,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
guard: {T},
ss-eq: x ≡ y,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
let: let,
top: Top,
rev_implies: P ⇐ Q,
rsub: x - y,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
le: A ≤ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
req: x = y,
rge: x ≥ y
Lemmas referenced :
ip-dist-between,
req_witness,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
ip-congruent_wf,
ip-between_wf,
ss-sep_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-point_wf,
stable_req,
false_wf,
or_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ip-between-iff,
ss-sep-symmetry,
ip-five-segment-lemma,
member_rooint_lemma,
radd-preserves-rless,
rsub_wf,
radd_wf,
rminus_wf,
rless_wf,
rless_functionality,
radd-rminus-assoc,
req_weakening,
radd_functionality,
radd_comm,
radd-int,
rnexp-req-iff,
less_than_wf,
rv-norm-nonneg,
rnexp_wf,
le_wf,
req_functionality,
rnexp_functionality,
rv-norm-sub,
rdiv_wf,
rmul_functionality,
ip-dist-between-2,
rv-add_wf,
rv-mul_wf,
rabs_wf,
rv-norm_functionality,
rv-sub_functionality,
ss-eq_inversion,
ss-eq_weakening,
rv-norm-positive,
rv-sep-iff,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
radd-zero-both,
equal_wf,
rmul_preserves_req,
req_inversion,
req_transitivity,
rleq_weakening_rless,
rabs-of-nonneg,
rdiv_functionality,
rsub_functionality,
ip-congruent-same2,
ip-congruent_functionality,
ip-congruent-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
promote_hyp,
sqequalRule,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
functionEquality,
lambdaFormation,
unionElimination,
voidElimination,
dependent_functionElimination,
productElimination,
voidEquality,
addEquality,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
inlFormation,
inrFormation
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c,d,A,B,C,D:Point].
(cd=CD) supposing (bd=BD and ad=AD and bc=BC and ab=AB and A\_B\_C and a\_b\_c and a \# b)
Date html generated:
2017_10_05-AM-00_03_24
Last ObjectModification:
2017_03_11-PM-06_35_32
Theory : inner!product!spaces
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