Nuprl Lemma : anti_sym_shift
∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ]. ∀[f:A ⟶ B].
(AntiSym(A;x,y.R[x;y])) supposing (AntiSym(B;x,y.S[x;y]) and RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f) and Inj(A;B;f))
Proof
Definitions occuring in Statement :
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f),
anti_sym: AntiSym(T;x,y.R[x; y]),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
anti_sym: AntiSym(T;x,y.R[x; y]),
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f),
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
anti_sym_wf,
rels_iso_wf,
inject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
hypothesis,
applyEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
thin,
axiomEquality,
universeEquality,
because_Cache,
lemma_by_obid,
isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
independent_functionElimination,
productElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}[S:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}]. \mforall{}[f:A {}\mrightarrow{} B].
(AntiSym(A;x,y.R[x;y])) supposing
(AntiSym(B;x,y.S[x;y]) and
RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f) and
Inj(A;B;f))
Date html generated:
2016_05_15-PM-00_03_36
Last ObjectModification:
2015_12_26-PM-11_25_04
Theory : gen_algebra_1
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