Nuprl Lemma : matrix-times-req-real-matrix-times
∀[n,a,b:ℕ]. ∀[A:ℝ(a × n)]. ∀[B:ℝ(n × b)]. (A*B) ≡ (A*B)
Proof
Definitions occuring in Statement :
real-matrix-times: (A*B),
reqmatrix: X ≡ Y,
rmatrix: ℝ(a × b),
real-ring: real-ring(),
nat: ℕ,
uall: ∀[x:A]. B[x],
matrix-times: (M*N)
Definitions unfolded in proof :
real-matrix-times: (A*B),
matrix-times: (M*N),
real-ring: real-ring(),
rng_times: *,
pi2: snd(t),
pi1: fst(t),
infix_ap: x f y,
matrix-ap: M[i,j],
rng_sum: rng_sum,
mx: matrix(M[x; y]),
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
grp_id: e,
rng_plus: +r,
rng_zero: 0,
reqmatrix: X ≡ Y,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
so_lambda: λ2x.t[x],
rmatrix: ℝ(a × b),
so_apply: x[s],
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
implies: P ⇒ Q
Lemmas referenced :
rsum-as-itop,
rmul_wf,
int_seg_wf,
itop_wf,
real_wf,
radd_wf,
int-to-real_wf,
req_witness,
real-matrix-times_wf,
subtype_rel_self,
rmatrix_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
lambdaEquality_alt,
applyEquality,
universeIsType,
because_Cache,
natural_numberEquality,
inhabitedIsType,
isect_memberFormation_alt,
dependent_functionElimination,
functionEquality,
imageElimination,
independent_functionElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[n,a,b:\mBbbN{}]. \mforall{}[A:\mBbbR{}(a \mtimes{} n)]. \mforall{}[B:\mBbbR{}(n \mtimes{} b)]. (A*B) \mequiv{} (A*B)
Date html generated:
2019_10_30-AM-08_16_41
Last ObjectModification:
2019_09_19-AM-11_36_33
Theory : reals
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