Nuprl Lemma : weak-antecedent-surjections-compose
∀es:EO
∀[P,Q,R:E ⟶ ℙ].
∀f:{e:E| P e} ⟶ {e:E| Q e} . ∀g:{e:E| Q e} ⟶ {e:E| R e} . ((Q ←←= f== P ∧ R ←←= g== Q) ⇒ R ←←= g o f== P)
Proof
Definitions occuring in Statement :
weak-antecedent-surjection: Q ←←= f== P,
es-E: E,
event_ordering: EO,
compose: f o g,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
weak-antecedent-surjection: Q ←←= f== P,
cand: A c∧ B,
exists: ∃x:A. B[x],
compose: f o g,
guard: {T},
weak-antecedent-function: Q ←==f== P
Latex:
\mforall{}es:EO
\mforall{}[P,Q,R:E {}\mrightarrow{} \mBbbP{}].
\mforall{}f:\{e:E| P e\} {}\mrightarrow{} \{e:E| Q e\} . \mforall{}g:\{e:E| Q e\} {}\mrightarrow{} \{e:E| R e\} .
((Q \mleftarrow{}\mleftarrow{}= f== P \mwedge{} R \mleftarrow{}\mleftarrow{}= g== Q) {}\mRightarrow{} R \mleftarrow{}\mleftarrow{}= g o f== P)
Date html generated:
2016_05_16-AM-10_17_39
Last ObjectModification:
2015_12_28-PM-09_23_33
Theory : new!event-ordering
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