Nuprl Lemma : det-mul-row
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[i:ℕn]. ∀[k:|r|]. (|matrix-mul-row(r;k;i;M)| = (k * |M|) ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-mul-row: matrix-mul-row(r;k;i;M),
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
ycomb: Y,
itop: Π(op,id) lb ≤ i < ub. E[i],
mon_itop: Π lb ≤ i < ub. E[i],
rng_prod: rng_prod,
let: let,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
so_apply: x[s],
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
injection: A →⟶ B,
rng: Rng,
so_lambda: λ2x.t[x],
nat: ℕ,
crng: CRng,
prop: ℙ,
squash: ↓T,
matrix-det: |M|,
member: t ∈ T,
uall: ∀[x:A]. B[x],
pi1: fst(t),
grp_car: |g|,
mul_mon_of_rng: r↓xmn,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
lelt: i ≤ j < k,
ge: i ≥ j ,
int_seg: {i..j-},
infix_ap: x f y,
mx: matrix(M[x; y]),
matrix-mul-row: matrix-mul-row(r;k;i;M),
matrix-ap: M[i,j],
pi2: snd(t),
grp_op: *
Lemmas referenced :
crng_wf,
nat_wf,
matrix_wf,
rng_times_over_minus,
rng_times_wf,
infix_ap_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
rng_wf,
iff_weakening_equal,
subtype_rel_list,
subtype_rel_set,
rng_times_lsum_l,
l_member_wf,
all_wf,
no_repeats_wf,
list_wf,
permutations-list_wf,
matrix-mul-row_wf,
matrix-ap_wf,
rng_prod_wf,
rng_minus_wf,
int_subtype_base,
equal-wf-base,
permutation-sign_wf,
rng_car_wf,
let_wf,
int_seg_wf,
injection_wf,
rng_lsum_wf,
true_wf,
squash_wf,
equal_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
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,
nat_properties,
int_seg_properties,
mul_mon_of_rng_wf_c,
mon_itop_split_el,
mul_mon_of_rng_wf,
grp_car_wf,
grp_op_wf,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
mon_itop_wf,
rng_times_assoc,
crng_times_ac_1,
crng_times_comm
Rules used in proof :
axiomEquality,
isect_memberEquality,
int_eqReduceFalseSq,
voidElimination,
instantiate,
promote_hyp,
dependent_pairFormation,
int_eqReduceTrueSq,
equalityElimination,
unionElimination,
lambdaFormation,
cumulativity,
independent_functionElimination,
productElimination,
imageMemberEquality,
functionExtensionality,
independent_isectElimination,
functionEquality,
dependent_functionElimination,
productEquality,
baseClosed,
closedConclusion,
baseApply,
intEquality,
setEquality,
int_eqEquality,
sqequalRule,
natural_numberEquality,
rename,
setElimination,
because_Cache,
universeEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
voidEquality,
approximateComputation,
applyLambdaEquality,
hyp_replacement,
addEquality,
dependent_set_memberEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[i:\mBbbN{}n]. \mforall{}[k:|r|]. (|matrix-mul-row(r;k;i;M)| = (k * |M|))
Date html generated:
2018_05_21-PM-09_36_28
Last ObjectModification:
2017_12_12-PM-03_52_10
Theory : matrices
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