Nuprl Lemma : scalar-triple-product-symmetry
∀[r:CRng]. ∀[a,b,c:ℕ3 ⟶ |r|]. (|a,b,c| = |b,c,a| ∈ |r|)
Proof
Definitions occuring in Statement :
scalar-triple-product: |a,b,c|,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
true: True,
squash: ↓T,
less_than: a < b,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
rng: Rng,
crng: CRng,
bfalse: ff,
cons: [a / b],
select: L[n],
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
lt_int: i <z j,
ycomb: Y,
itop: Π(op,id) lb ≤ i < ub. E[i],
grp_id: e,
pi1: fst(t),
pi2: snd(t),
grp_op: *,
add_grp_of_rng: r↓+gp,
mon_itop: Π lb ≤ i < ub. E[i],
rng_sum: rng_sum,
scalar-product: (a . b),
cross-product: (a x b),
scalar-triple-product: |a,b,c|,
member: t ∈ T,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rng_minus_wf,
rng_times_wf,
infix_ap_wf,
lelt_wf,
false_wf,
rng_zero_wf,
rng_plus_wf,
crng_wf,
rng_car_wf,
int_seg_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_over_plus,
rng_times_over_minus,
crng_times_comm,
crng_times_ac_1,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_comm,
rng_plus_zero,
iff_weakening_equal
Rules used in proof :
baseClosed,
imageMemberEquality,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
because_Cache,
axiomEquality,
isect_memberEquality,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
functionEquality,
hypothesis,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[a,b,c:\mBbbN{}3 {}\mrightarrow{} |r|]. (|a,b,c| = |b,c,a|)
Date html generated:
2018_05_21-PM-09_45_10
Last ObjectModification:
2017_12_18-PM-01_26_25
Theory : matrices
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