Nuprl Lemma : cross-product_wf
∀[r:RngSig]. ∀[a,b:ℕ3 ⟶ |r|]. ((a x b) ∈ ℕ3 ⟶ |r|)
Proof
Definitions occuring in Statement :
cross-product: (a x b),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
guard: {T},
uimplies: b supposing a,
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-},
cross-product: (a x b),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_sig_wf,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
decidable__lt,
length_of_nil_lemma,
length_of_cons_lemma,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
nil_wf,
rng_minus_wf,
lelt_wf,
false_wf,
int_seg_wf,
rng_times_wf,
rng_plus_wf,
infix_ap_wf,
cons_wf,
rng_car_wf,
select_wf
Rules used in proof :
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
addEquality,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
unionElimination,
dependent_functionElimination,
productElimination,
independent_isectElimination,
rename,
setElimination,
baseClosed,
imageMemberEquality,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
functionExtensionality,
applyEquality,
because_Cache,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:RngSig]. \mforall{}[a,b:\mBbbN{}3 {}\mrightarrow{} |r|]. ((a x b) \mmember{} \mBbbN{}3 {}\mrightarrow{} |r|)
Date html generated:
2018_05_21-PM-09_40_56
Last ObjectModification:
2017_12_18-PM-00_07_58
Theory : matrices
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