Nuprl Lemma : det-id
∀[r:CRng]. ∀[n:ℕ]. (|I| = 1 ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
identity-matrix: I,
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
deq: EqDecider(T),
prop: ℙ,
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
injection: A →⟶ B,
true: True,
guard: {T},
infix_ap: x f y,
istype: istype(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
sq_stable: SqStable(P),
l_member: (x ∈ l),
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q),
l_all: (∀x∈L.P[x]),
matrix-det: |M|,
identity-matrix: I,
matrix-ap: M[i,j],
mx: matrix(M[x; y]),
let: let,
nequal: a ≠ b ∈ T ,
rng_prod: rng_prod,
rng_car: |r|,
pi1: fst(t),
grp_car: |g|,
mul_mon_of_rng: r↓xmn,
grp_op: *,
pi2: snd(t)
Lemmas referenced :
deq-exists,
int_seg_wf,
decidable__equal_compact_domain,
decidable__equal_int_seg,
compact-finite,
istype-nat,
crng_wf,
rng_lsum-split,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rng_car_wf,
ifthenelse_wf,
rng_one_wf,
rng_zero_wf,
permutations-list_wf,
subtype_rel_set,
list_wf,
injection_wf,
no_repeats_wf,
l_member_wf,
subtype_rel_list,
subtype_rel_self,
iff_weakening_equal,
rng_plus_wf,
rng_lsum_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
filter_type,
bnot_wf,
rng_wf,
assert_wf,
istype-assert,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
equal-wf-T-base,
eqof_wf,
not_wf,
equal-wf-base-T,
istype-void,
iff_transitivity,
assert_of_bnot,
rng_lsum_0,
filter_is_singleton,
no_repeats-l_member!,
sq_stable__no_repeats,
no_repeats-strong-subtype,
strong-subtype-set3,
inject_wf,
strong-subtype-self,
sq_stable__and,
sq_stable__all,
sq_stable__l_member,
decidable__equal_injection,
identity-injection,
respects-equality-set-trivial2,
change-equality-type,
istype-less_than,
length_wf,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
int_seg_properties,
decidable__lt,
istype-le,
intformless_wf,
int_formula_prop_less_lemma,
strong-subtype-deq-subtype,
rng_lsum_cons_lemma,
rng_lsum_nil_lemma,
equal-wf-base,
rng_plus_assoc,
rng_plus_zero,
assert-deq,
rng_minus_wf,
rng_prod_wf,
let_wf,
permutation-sign_wf,
int_subtype_base,
rng_prod_1,
permutation-sign-id,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__exists_int_seg,
set_subtype_base,
lelt_wf,
decidable__not,
decidable__equal_int,
mon_itop_split_el,
mul_mon_of_rng_wf_c,
grp_car_wf,
mul_mon_of_rng_wf,
rng_times_wf,
itermAdd_wf,
int_term_value_add_lemma,
rng_times_zero,
rng_minus_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
imageElimination,
lambdaFormation_alt,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
inhabitedIsType,
universeIsType,
functionIsType,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
applyEquality,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
productEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
setEquality,
setIsType,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
cumulativity,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
productIsType,
closedConclusion,
approximateComputation,
int_eqEquality,
sqequalBase,
int_eqReduceTrueSq,
functionExtensionality,
baseApply,
intEquality,
int_eqReduceFalseSq,
addEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. (|I| = 1)
Date html generated:
2019_10_16-AM-11_27_22
Last ObjectModification:
2019_06_25-PM-03_26_38
Theory : matrices
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