Nuprl Lemma : injection-if-compose-injection
∀[T:Type]. ∀[f,g:T ⟶ T]. Inj(T;T;f) supposing Inj(T;T;g o f)
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
compose: f o g,
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,
compose: f o g,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
inject_wf,
compose_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
lambdaFormation,
hypothesis,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[f,g:T {}\mrightarrow{} T]. Inj(T;T;f) supposing Inj(T;T;g o f)
Date html generated:
2017_04_14-AM-07_34_15
Last ObjectModification:
2017_02_27-PM-03_07_27
Theory : fun_1
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