![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sitmfval | Structured version Visualization version GIF version |
Description: Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
Ref | Expression |
---|---|
sitmval.d | ⊢ 𝐷 = (dist‘𝑊) |
sitmval.1 | ⊢ (𝜑 → 𝑊 ∈ 𝑉) |
sitmval.2 | ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) |
sitmfval.1 | ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) |
sitmfval.2 | ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) |
Ref | Expression |
---|---|
sitmfval | ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sitmval.d | . . 3 ⊢ 𝐷 = (dist‘𝑊) | |
2 | sitmval.1 | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑉) | |
3 | sitmval.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
4 | 1, 2, 3 | sitmval 33337 | . 2 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔)))) |
5 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹) | |
6 | simprr 772 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺) | |
7 | 5, 6 | oveq12d 7424 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 ∘f 𝐷𝑔) = (𝐹 ∘f 𝐷𝐺)) |
8 | 7 | fveq2d 6893 | . 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔)) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺))) |
9 | sitmfval.1 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) | |
10 | sitmfval.2 | . 2 ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) | |
11 | fvexd 6904 | . 2 ⊢ (𝜑 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺)) ∈ V) | |
12 | 4, 8, 9, 10, 11 | ovmpod 7557 | 1 ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cuni 4908 dom cdm 5676 ran crn 5677 ‘cfv 6541 (class class class)co 7406 ∘f cof 7665 0cc0 11107 +∞cpnf 11242 [,]cicc 13324 ↾s cress 17170 distcds 17203 ℝ*𝑠cxrs 17443 measurescmeas 33182 sitmcsitm 33316 sitgcsitg 33317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-1st 7972 df-2nd 7973 df-sitm 33319 |
This theorem is referenced by: sitmcl 33339 sitmf 33340 |
Copyright terms: Public domain | W3C validator |