Nuprl Lemma : ip-triangle-permute
∀rv:InnerProductSpace. ∀a,b,c:Point(rv). (Δ(a;b;c) ⇒ Δ(c;a;b))
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
inner-product-space: InnerProductSpace,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ip-triangle: Δ(a;b;c),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
rv-sub: x - y,
rv-minus: -x,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
true: True,
absval: |i|,
squash: ↓T,
false: False,
not: ¬A
Lemmas referenced :
ip-triangle-permute-lemma,
rv-sub_wf,
inner-product-space_subtype,
ip-triangle_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
Error :ss-eq_wf,
rv-add_wf,
rv-mul_wf,
int-to-real_wf,
radd_wf,
rmul_wf,
rv-minus_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermAdd_wf,
rv-0_wf,
rabs_wf,
rv-ip_wf,
rv-norm_wf,
uiff_transitivity,
Error :ss-eq_functionality,
rv-add_functionality,
Error :ss-eq_weakening,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
Error :ss-eq_transitivity,
rv-add-swap,
rv-add-comm,
rv-mul-add-alt,
rv-mul_functionality,
req_transitivity,
radd_functionality,
req_weakening,
rv-mul0,
rv-0-add,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
rless_functionality,
rabs_functionality,
rv-ip_functionality,
rmul_functionality,
rv-norm_functionality,
rv-norm-difference-symmetry,
rmul_comm,
rv-mul1,
req_functionality,
req_wf,
absval_wf,
uiff_transitivity2,
rv-ip-mul2,
rabs-rmul,
squash_wf,
true_wf,
real_wf,
rabs-int,
rv-ip-symmetry,
itermVar_wf,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_functionElimination,
universeIsType,
inhabitedIsType,
instantiate,
independent_isectElimination,
minusEquality,
natural_numberEquality,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
productElimination,
approximateComputation,
isect_memberEquality_alt,
voidElimination,
equalityIstype,
imageElimination,
imageMemberEquality,
baseClosed,
int_eqEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point(rv). (\mDelta{}(a;b;c) {}\mRightarrow{} \mDelta{}(c;a;b))
Date html generated:
2020_05_20-PM-01_13_25
Last ObjectModification:
2019_12_10-AM-00_41_54
Theory : inner!product!spaces
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