-
Notifications
You must be signed in to change notification settings - Fork 0
/
Imagen_de_la_interseccion_general.thy
68 lines (59 loc) · 1.91 KB
/
Imagen_de_la_interseccion_general.thy
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
(* Imagen_de_la_interseccion_general.thy
-- Imagen de la intersección general
-- José A. Alonso Jiménez
-- Sevilla, 24 de junio de 2021
-- ------------------------------------------------------------------ *)
(* ---------------------------------------------------------------------
-- Demostrar que
-- f ` (\<Inter> i, A i) \<subseteq> \<Inter> i, f ` A i
-- ------------------------------------------------------------------ *)
theory Imagen_de_la_interseccion_general
imports Main
begin
(* 1\<ordfeminine> demostración *)
lemma "f ` (\<Inter> i \<in> I. A i) \<subseteq> (\<Inter> i \<in> I. f ` A i)"
proof (rule subsetI)
fix y
assume "y \<in> f ` (\<Inter> i \<in> I. A i)"
then show "y \<in> (\<Inter> i \<in> I. f ` A i)"
proof (rule imageE)
fix x
assume "y = f x"
assume xIA : "x \<in> (\<Inter> i \<in> I. A i)"
have "f x \<in> (\<Inter> i \<in> I. f ` A i)"
proof (rule INT_I)
fix i
assume "i \<in> I"
with xIA have "x \<in> A i"
by (rule INT_D)
then show "f x \<in> f ` A i"
by (rule imageI)
qed
with \<open>y = f x\<close> show "y \<in> (\<Inter> i \<in> I. f ` A i)"
by (rule ssubst)
qed
qed
(* 2\<ordfeminine> demostración *)
lemma "f ` (\<Inter> i \<in> I. A i) \<subseteq> (\<Inter> i \<in> I. f ` A i)"
proof
fix y
assume "y \<in> f ` (\<Inter> i \<in> I. A i)"
then show "y \<in> (\<Inter> i \<in> I. f ` A i)"
proof
fix x
assume "y = f x"
assume xIA : "x \<in> (\<Inter> i \<in> I. A i)"
have "f x \<in> (\<Inter> i \<in> I. f ` A i)"
proof
fix i
assume "i \<in> I"
with xIA have "x \<in> A i" by simp
then show "f x \<in> f ` A i" by simp
qed
with \<open>y = f x\<close> show "y \<in> (\<Inter> i \<in> I. f ` A i)" by simp
qed
qed
(* 3\<ordfeminine> demostración *)
lemma "f ` (\<Inter> i \<in> I. A i) \<subseteq> (\<Inter> i \<in> I. f ` A i)"
by auto
end