Nuprl Lemma : compose-surjections
∀[A,B,C:Type]. ∀[f:A ⟶ B]. ∀[g:B ⟶ C]. (Surj(A;B;f) ⇒ Surj(B;C;g) ⇒ Surj(A;C;g o f))
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
compose: f o g,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
and: P ∧ Q,
compose: f o g,
prop: ℙ,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
surject: Surj(A;B;f),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
equal_wf,
istype-universe,
surject_wf,
compose_wf
Rules used in proof :
rename,
setElimination,
applyLambdaEquality,
productIsType,
independent_pairFormation,
dependent_set_memberEquality_alt,
sqequalRule,
hyp_replacement,
universeEquality,
instantiate,
functionIsType,
universeIsType,
isectElimination,
extract_by_obid,
introduction,
cut,
applyEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
inhabitedIsType,
equalityIstype,
dependent_pairFormation_alt,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[g:B {}\mrightarrow{} C]. (Surj(A;B;f) {}\mRightarrow{} Surj(B;C;g) {}\mRightarrow{} Surj(A;C;g o f))
Date html generated:
2019_10_15-AM-10_20_34
Last ObjectModification:
2019_10_08-PM-00_18_08
Theory : fun_1
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