Metamath Proof Explorer


Theorem prdsds

Description: Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by Zhi Wang, 18-Aug-2024)

Ref Expression
Hypotheses prdsbas.p P = S 𝑠 R
prdsbas.s φ S V
prdsbas.r φ R W
prdsbas.b B = Base P
prdsbas.i φ dom R = I
prdsds.l D = dist P
Assertion prdsds φ D = f B , g B sup ran x I f x dist R x g x 0 * <

Proof

Step Hyp Ref Expression
1 prdsbas.p P = S 𝑠 R
2 prdsbas.s φ S V
3 prdsbas.r φ R W
4 prdsbas.b B = Base P
5 prdsbas.i φ dom R = I
6 prdsds.l D = dist P
7 eqid Base S = Base S
8 1 2 3 4 5 prdsbas φ B = x I Base R x
9 eqid + P = + P
10 1 2 3 4 5 9 prdsplusg φ + P = f B , g B x I f x + R x g x
11 eqid P = P
12 1 2 3 4 5 11 prdsmulr φ P = f B , g B x I f x R x g x
13 eqid P = P
14 1 2 3 4 5 7 13 prdsvsca φ P = f Base S , g B x I f R x g x
15 eqidd φ f B , g B S x I f x 𝑖 R x g x = f B , g B S x I f x 𝑖 R x g x
16 eqidd φ 𝑡 TopOpen R = 𝑡 TopOpen R
17 eqid P = P
18 1 2 3 4 5 17 prdsle φ P = f g | f g B x I f x R x g x
19 eqidd φ f B , g B sup ran x I f x dist R x g x 0 * < = f B , g B sup ran x I f x dist R x g x 0 * <
20 eqidd φ f B , g B x I f x Hom R x g x = f B , g B x I f x Hom R x g x
21 eqidd φ a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x = a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x
22 1 7 5 8 10 12 14 15 16 18 19 20 21 2 3 prdsval φ P = Base ndx B + ndx + P ndx P Scalar ndx S ndx P 𝑖 ndx f B , g B S x I f x 𝑖 R x g x TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * < Hom ndx f B , g B x I f x Hom R x g x comp ndx a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x
23 dsid dist = Slot dist ndx
24 4 fvexi B V
25 xrex * V
26 25 uniex * V
27 26 pwex 𝒫 * V
28 df-sup sup ran x I f x dist R x g x 0 * < = y * | z ran x I f x dist R x g x 0 ¬ y < z z * z < y w ran x I f x dist R x g x 0 z < w
29 ssrab2 y * | z ran x I f x dist R x g x 0 ¬ y < z z * z < y w ran x I f x dist R x g x 0 z < w *
30 29 unissi y * | z ran x I f x dist R x g x 0 ¬ y < z z * z < y w ran x I f x dist R x g x 0 z < w *
31 26 30 elpwi2 y * | z ran x I f x dist R x g x 0 ¬ y < z z * z < y w ran x I f x dist R x g x 0 z < w 𝒫 *
32 28 31 eqeltri sup ran x I f x dist R x g x 0 * < 𝒫 *
33 32 rgen2w f B g B sup ran x I f x dist R x g x 0 * < 𝒫 *
34 24 24 27 33 mpoexw f B , g B sup ran x I f x dist R x g x 0 * < V
35 34 a1i φ f B , g B sup ran x I f x dist R x g x 0 * < V
36 snsstp3 dist ndx f B , g B sup ran x I f x dist R x g x 0 * < TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * <
37 ssun1 TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * < TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * < Hom ndx f B , g B x I f x Hom R x g x comp ndx a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x
38 36 37 sstri dist ndx f B , g B sup ran x I f x dist R x g x 0 * < TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * < Hom ndx f B , g B x I f x Hom R x g x comp ndx a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x
39 ssun2 TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * < Hom ndx f B , g B x I f x Hom R x g x comp ndx a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x Base ndx B + ndx + P ndx P Scalar ndx S ndx P 𝑖 ndx f B , g B S x I f x 𝑖 R x g x TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * < Hom ndx f B , g B x I f x Hom R x g x comp ndx a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x
40 38 39 sstri dist ndx f B , g B sup ran x I f x dist R x g x 0 * < Base ndx B + ndx + P ndx P Scalar ndx S ndx P 𝑖 ndx f B , g B S x I f x 𝑖 R x g x TopSet ndx 𝑡 TopOpen R ndx P dist ndx f B , g B sup ran x I f x dist R x g x 0 * < Hom ndx f B , g B x I f x Hom R x g x comp ndx a B × B , c B d 2 nd a f B , g B x I f x Hom R x g x c , e f B , g B x I f x Hom R x g x a x I d x 1 st a x 2 nd a x comp R x c x e x
41 22 6 23 35 40 prdsbaslem φ D = f B , g B sup ran x I f x dist R x g x 0 * <