Nuprl Lemma : cross-product-distrib1
∀[r:CRng]. ∀[a,b,c:ℕ3 ⟶ |r|]. ((a x (b + c)) = ((a x b) + (a x c)) ∈ (ℕ3 ⟶ |r|))
Proof
Definitions occuring in Statement :
cross-product: (a x b),
vector-add: (a + b),
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,
rng: Rng,
crng: CRng,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
subtract: n - m,
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
cons: [a / b],
select: L[n],
vector-add: (a + b),
cross-product: (a x b),
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
int_seg: {i..j-},
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rng_times_wf,
infix_ap_wf,
rng_minus_wf,
lelt_wf,
rng_plus_wf,
crng_wf,
rng_car_wf,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_cases,
false_wf,
int_seg_subtype,
int_seg_properties,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
equal_wf,
squash_wf,
true_wf,
rng_times_over_plus,
rng_minus_over_plus,
rng_plus_assoc,
rng_plus_comm,
rng_plus_ac_1,
iff_weakening_equal
Rules used in proof :
baseClosed,
imageMemberEquality,
dependent_set_memberEquality,
applyEquality,
axiomEquality,
functionEquality,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
approximateComputation,
productElimination,
lambdaFormation,
independent_pairFormation,
addEquality,
hypothesis_subsumption,
sqequalRule,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
because_Cache,
independent_isectElimination,
intEquality,
cumulativity,
isectElimination,
instantiate,
unionElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
functionExtensionality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
universeEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[a,b,c:\mBbbN{}3 {}\mrightarrow{} |r|]. ((a x (b + c)) = ((a x b) + (a x c)))
Date html generated:
2018_05_21-PM-09_41_12
Last ObjectModification:
2017_12_18-PM-00_33_50
Theory : matrices
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