Nuprl Lemma : real-matrix-scalar-mul_functionality
∀[a,b:ℕ]. ∀[c1,c2:ℝ]. ∀[A,B:ℝ(a × b)]. (c1*A ≡ c2*B) supposing ((c1 = c2) and A ≡ B)
Proof
Definitions occuring in Statement :
real-matrix-scalar-mul: c*A,
reqmatrix: X ≡ Y,
rmatrix: ℝ(a × b),
req: x = y,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
reqmatrix: X ≡ Y,
rmatrix: ℝ(a × b),
real-matrix-scalar-mul: c*A,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_wf,
req_witness,
real-matrix-scalar-mul_wf,
subtype_rel_self,
real_wf,
req_wf,
reqmatrix_wf,
rmatrix_wf,
istype-nat,
rmul_wf,
req_weakening,
req_functionality,
rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
functionEquality,
imageElimination,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
independent_isectElimination
Latex:
\mforall{}[a,b:\mBbbN{}]. \mforall{}[c1,c2:\mBbbR{}]. \mforall{}[A,B:\mBbbR{}(a \mtimes{} b)]. (c1*A \mequiv{} c2*B) supposing ((c1 = c2) and A \mequiv{} B)
Date html generated:
2019_10_30-AM-08_19_41
Last ObjectModification:
2019_09_19-PM-01_02_14
Theory : reals
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