Nuprl Lemma : usym_shift
∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ].
∀f:A ⟶ B
((∀[x,y:A]. R[x;y] supposing R[x;y])
⇒ RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)
⇒ UniformlySym(B;x,y.S[x;y])
⇒ UniformlySym(A;x,y.R[x;y]))
Proof
Definitions occuring in Statement :
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)
,
usym: UniformlySym(T;x,y.E[x; y])
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
so_apply: x[s1;s2]
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
function: x:A ⟶ B[x]
,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
usym: UniformlySym(T;x,y.E[x; y])
,
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)
,
member: t ∈ T
,
prop: ℙ
,
so_apply: x[s1;s2]
,
so_lambda: λ2x y.t[x; y]
,
so_lambda: λ2x.t[x]
,
uimplies: b supposing a
,
subtype_rel: A ⊆r B
,
so_apply: x[s]
,
sq_stable: SqStable(P)
,
iff: P
⇐⇒ Q
,
and: P ∧ Q
,
squash: ↓T
,
rev_implies: P
⇐ Q
,
guard: {T}
Lemmas referenced :
usym_wf,
rels_iso_wf,
uall_wf,
isect_wf,
subtype_rel_self,
uimplies-iff-squash-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesis,
instantiate,
universeEquality,
because_Cache,
functionEquality,
cumulativity,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}[S:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
\mforall{}f:A {}\mrightarrow{} B
((\mforall{}[x,y:A]. R[x;y] supposing R[x;y])
{}\mRightarrow{} RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)
{}\mRightarrow{} UniformlySym(B;x,y.S[x;y])
{}\mRightarrow{} UniformlySym(A;x,y.R[x;y]))
Date html generated:
2019_10_15-AM-10_32_24
Last ObjectModification:
2018_08_25-PM-05_13_09
Theory : gen_algebra_1
Home
Index