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two lemmas from infotheo #406

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Jul 14, 2021
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two lemmas from infotheo #406

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2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -8,6 +8,8 @@
+ lemmas `seriesN`, `seriesD`, `seriesZ`, `is_cvg_seriesN`, `lim_seriesN`,
`is_cvg_seriesZ`, `lim_seriesZ`, `is_cvg_seriesD`, `lim_seriesD`,
`is_cvg_seriesB`, `lim_seriesB`, `lim_series_le`, `lim_series_norm`
- in `classical_sets.v`:
+ lemmas `bigcup_image`, `bigcup_of_set1`

### Changed

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19 changes: 16 additions & 3 deletions theories/classical_sets.v
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Expand Up @@ -851,14 +851,27 @@ Qed.

End bigop_nat_lemmas.

Lemma preimage_bigcup {aT rT I} (P : set I) (f : aT -> rT) A :
f @^-1` (\bigcup_ (i in P) A i) = \bigcup_(i in P) (f @^-1` A i).
Lemma preimage_bigcup {aT rT I} (P : set I) (f : aT -> rT) (F : I -> set rT) :
f @^-1` (\bigcup_ (i in P) F i) = \bigcup_(i in P) (f @^-1` F i).
Proof. exact/predeqP. Qed.

Lemma preimage_bigcap {aT rT I} (P : set I) (f : aT -> rT) F :
Lemma preimage_bigcap {aT rT I} (P : set I) (f : aT -> rT) (F : I -> set rT) :
f @^-1` (\bigcap_ (i in P) F i) = \bigcap_(i in P) (f @^-1` F i).
Proof. exact/predeqP. Qed.

Lemma bigcup_image {aT rT I} (P : set aT) (f : aT -> I) (F : I -> set rT) :
\bigcup_(x in f @` P) F x = \bigcup_(x in P) F (f x).
Proof.
rewrite eqEsubset; split=> x; first by case=> j [] i pi <- Xfix; exists i.
by case=> i Pi Ffix; exists (f i); [exists i|].
Qed.

Lemma bigcup_of_set1 {rT I} (P : set I) (f : I -> rT) :
\bigcup_(x in P) [set f x] = f @` P.
Proof.
by rewrite eqEsubset; split=>[a [i ?]->| a [i ?]<-]; [apply: imageP | exists i].
Qed.

Lemma setM0 T1 T2 (A1 : set T1) : A1 `*` set0 = set0 :> set (T1 * T2).
Proof. by rewrite predeqE => -[t u]; split => // -[]. Qed.

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