Nuprl Lemma : reqmatrix_functionality
∀[a,b:ℕ]. ∀[X1,X2,Y1,Y2:ℝ(a × b)]. (uiff(X1 ≡ Y1;X2 ≡ Y2)) supposing (Y1 ≡ Y2 and X1 ≡ X2)
Proof
Definitions occuring in Statement :
reqmatrix: X ≡ Y,
rmatrix: ℝ(a × b),
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
reqmatrix: X ≡ Y,
rmatrix: ℝ(a × b),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
nat: ℕ,
implies: P ⇒ Q,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_wf,
req_witness,
req_wf,
real_wf,
istype-nat,
req_weakening,
req_functionality,
req_transitivity,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
because_Cache,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_isectElimination
Latex:
\mforall{}[a,b:\mBbbN{}]. \mforall{}[X1,X2,Y1,Y2:\mBbbR{}(a \mtimes{} b)]. (uiff(X1 \mequiv{} Y1;X2 \mequiv{} Y2)) supposing (Y1 \mequiv{} Y2 and X1 \mequiv{} X2)
Date html generated:
2019_10_30-AM-08_14_57
Last ObjectModification:
2019_09_19-AM-11_04_40
Theory : reals
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