Nuprl Lemma : ip-between-inner-trans
∀[rv:InnerProductSpace]. ∀[a,b,c,d:Point(rv)]. (a_b_d ⇒ b_c_d ⇒ a_b_c)
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
ip-between: a_b_c,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
ss-eq: Error :ss-eq,
not: ¬A,
false: False,
stable: Stable{P},
uiff: uiff(P;Q),
top: Top,
cand: A c∧ B,
req_int_terms: t1 ≡ t2,
rneq: x ≠ y,
i-member: r ∈ I,
rooint: (l, u),
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
rat_term_to_real: rat_term_to_real(f;t),
rtermAdd: left "+" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermMultiply: left "*" right,
rtermSubtract: left "-" right,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
ip-between_wf,
req_witness,
radd_wf,
rmul_wf,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
rv-ip_wf,
int-to-real_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
stable__ip-between,
false_wf,
Error :ss-sep_wf,
not_wf,
ip-between-iff,
Error :ss-sep-symmetry,
ip-between-trivial2,
ip-between-trivial,
istype-void,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ip-between_functionality,
Error :ss-eq_weakening,
Error :ss-eq_inversion,
rv-add_wf,
rv-mul_wf,
rsub_wf,
Error :ss-eq_wf,
i-member_wf,
rooint_wf,
iff_weakening_uiff,
Error :ss-eq_functionality,
rv-add_functionality,
rv-mul_functionality,
req_weakening,
member_rooint_lemma,
rmul_preserves_rless,
radd-preserves-rless,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
rless-implies-rless,
req-iff-rsub-is-0,
rless_transitivity2,
rleq_weakening_rless,
itermAdd_wf,
rless_functionality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
rdiv_wf,
rless_wf,
rminus_wf,
rinv_wf2,
itermMinus_wf,
req_transitivity,
radd_functionality,
rminus_functionality,
rmul-rinv3,
real_term_value_minus_lemma,
rmul_preserves_req,
req_functionality,
rmul_functionality,
rmul-rinv,
req_wf,
squash_wf,
true_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
rv-mul-linear,
rv-add-assoc,
uiff_transitivity,
rv-mul-mul,
rv-mul-add-alt,
rv-mul-add,
assert-rat-term-eq2,
rtermVar_wf,
rtermAdd_wf,
rtermDivide_wf,
rtermSubtract_wf,
rtermMultiply_wf,
rtermConstant_wf,
ip-between-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
independent_functionElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
independent_isectElimination,
unionEquality,
functionEquality,
productElimination,
unionIsType,
functionIsType,
unionElimination,
voidElimination,
productIsType,
dependent_pairFormation_alt,
independent_pairFormation,
promote_hyp,
approximateComputation,
int_eqEquality,
closedConclusion,
inrFormation_alt,
equalityIstype,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c,d:Point(rv)]. (a\_b\_d {}\mRightarrow{} b\_c\_d {}\mRightarrow{} a\_b\_c)
Date html generated:
2020_05_20-PM-01_13_39
Last ObjectModification:
2019_12_08-PM-07_01_54
Theory : inner!product!spaces
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