Nuprl Lemma : cancel_shift
∀[A,B:Type]. ∀[opa:A ⟶ A ⟶ A]. ∀[opb:B ⟶ B ⟶ B]. ∀[f:A ⟶ B].
(Cancel(A;A;opa)) supposing (Cancel(B;B;opb) and FunThru2op(A;B;opa;opb;f) and Inj(A;B;f))
Proof
Definitions occuring in Statement :
cancel: Cancel(T;S;op),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cancel: Cancel(T;S;op),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
inject: Inj(A;B;f),
prop: ℙ,
infix_ap: x f y,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
equal_wf,
cancel_wf,
fun_thru_2op_wf,
inject_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[opa:A {}\mrightarrow{} A {}\mrightarrow{} A]. \mforall{}[opb:B {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[f:A {}\mrightarrow{} B].
(Cancel(A;A;opa)) supposing (Cancel(B;B;opb) and FunThru2op(A;B;opa;opb;f) and Inj(A;B;f))
Date html generated:
2017_10_01-AM-08_13_06
Last ObjectModification:
2017_02_28-PM-01_57_25
Theory : gen_algebra_1
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