Metamath Proof Explorer


Theorem sitmf

Description: The integral metric as a function. (Contributed by Thierry Arnoux, 13-Mar-2018)

Ref Expression
Hypotheses sitmf.0 φ W Mnd
sitmf.1 φ W ∞MetSp
sitmf.2 φ M ran measures
Assertion sitmf φ |. - .| M W : M W × M W 0 +∞

Proof

Step Hyp Ref Expression
1 sitmf.0 φ W Mnd
2 sitmf.1 φ W ∞MetSp
3 sitmf.2 φ M ran measures
4 eqid dist W = dist W
5 2 adantr φ f M W g M W W ∞MetSp
6 3 adantr φ f M W g M W M ran measures
7 simprl φ f M W g M W f M W
8 simprr φ f M W g M W g M W
9 4 5 6 7 8 sitmfval φ f M W g M W |g - f| M W = 𝑠 * 𝑠 0 +∞ f dist W f g dM
10 1 adantr φ f M W g M W W Mnd
11 10 5 6 7 8 sitmcl φ f M W g M W |g - f| M W 0 +∞
12 9 11 eqeltrrd φ f M W g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM 0 +∞
13 12 ralrimivva φ f M W g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM 0 +∞
14 eqid f M W , g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM = f M W , g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM
15 14 fmpo f M W g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM 0 +∞ f M W , g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM : M W × M W 0 +∞
16 13 15 sylib φ f M W , g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM : M W × M W 0 +∞
17 4 2 3 sitmval φ W sitm M = f M W , g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM
18 17 feq1d φ |. - .| M W : M W × M W 0 +∞ f M W , g M W 𝑠 * 𝑠 0 +∞ f dist W f g dM : M W × M W 0 +∞
19 16 18 mpbird φ |. - .| M W : M W × M W 0 +∞