Nuprl Lemma : matrix-ring_wf
∀[r:Rng]. ∀[n:ℕ]. (matrix-ring(r;n) ∈ Rng)
Proof
Definitions occuring in Statement :
matrix-ring: matrix-ring(r;n),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng: Rng
Definitions unfolded in proof :
prop: ℙ,
rng: Rng,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
rng_sig: RngSig,
matrix-ring: matrix-ring(r;n),
cand: A c∧ B,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
and: P ∧ Q,
inverse: Inverse(T;op;id;inv),
assoc: Assoc(T;op),
ident: Ident(T;op;id),
infix_ap: x f y,
group_p: IsGroup(T;op;id;inv),
monoid_p: IsMonoid(T;op;id),
bilinear: BiLinear(T;pl;tm),
rng_one: 1,
rng_times: *,
rng_minus: -r,
rng_zero: 0,
pi2: snd(t),
rng_plus: +r,
pi1: fst(t),
rng_car: |r|,
ring_p: IsRing(T;plus;zero;neg;times;one)
Lemmas referenced :
rng_wf,
nat_wf,
rng_one_wf,
rng_times_wf,
rng_minus_wf,
rng_zero_wf,
rng_plus_wf,
rng_car_wf,
ring_p_wf,
bool_wf,
unit_wf2,
it_wf,
identity-matrix_wf,
matrix-times_wf,
matrix-minus_wf,
zero-matrix_wf,
matrix-plus_wf,
bfalse_wf,
matrix_wf,
matrix-times-distrib-right,
matrix-times-distrib-left,
matrix-times-id-left,
matrix-times-id-right,
matrix-times-assoc,
matrix-plus-minus-left,
matrix-plus-minus-right,
matrix-plus-zero-left,
matrix-plus-zero-right,
iff_weakening_equal,
matrix-plus-assoc,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
because_Cache,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
dependent_set_memberEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cumulativity,
productEquality,
unionEquality,
functionEquality,
inrEquality,
lambdaEquality,
rename,
setElimination,
dependent_pairEquality,
independent_pairEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
imageElimination,
applyEquality,
independent_pairFormation
Latex:
\mforall{}[r:Rng]. \mforall{}[n:\mBbbN{}]. (matrix-ring(r;n) \mmember{} Rng)
Date html generated:
2018_05_21-PM-09_35_27
Last ObjectModification:
2017_12_11-PM-00_29_43
Theory : matrices
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