Nuprl Lemma : p-compose-inject
∀[A,B,C:Type]. ∀[g:A ⟶ (B + Top)]. ∀[f:B ⟶ (C + Top)].
(p-inject(A;C;f o g)) supposing (p-inject(B;C;f) and p-inject(A;B;g))
Proof
Definitions occuring in Statement :
p-inject: p-inject(A;B;f),
p-compose: f o g,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
p-inject: p-inject(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
guard: {T},
and: P ∧ Q
Lemmas referenced :
equal_wf,
do-apply_wf,
p-compose_wf,
assert_wf,
can-apply_wf,
top_wf,
subtype_rel_union,
p-inject_wf,
can-apply-compose,
do-apply-compose
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
hypothesis,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
independent_isectElimination,
because_Cache,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
unionEquality,
universeEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:A {}\mrightarrow{} (B + Top)]. \mforall{}[f:B {}\mrightarrow{} (C + Top)].
(p-inject(A;C;f o g)) supposing (p-inject(B;C;f) and p-inject(A;B;g))
Date html generated:
2018_05_21-PM-06_32_59
Last ObjectModification:
2017_07_26-PM-04_51_58
Theory : general
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