Mathbox for Thierry Arnoux |
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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 29738 | . 2 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘𝑓 𝐷𝑔)))) |
5 | simprl 790 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹) | |
6 | simprr 792 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺) | |
7 | 5, 6 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 ∘𝑓 𝐷𝑔) = (𝐹 ∘𝑓 𝐷𝐺)) |
8 | 7 | fveq2d 6107 | . 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 | fvex 6113 | . . 3 ⊢ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)) ∈ V | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)) ∈ V) |
13 | 4, 8, 9, 10, 12 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 0cc0 9815 +∞cpnf 9950 [,]cicc 12049 ↾s cress 15696 distcds 15777 ℝ*𝑠cxrs 15983 measurescmeas 29585 sitmcsitm 29717 sitgcsitg 29718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-1st 7059 df-2nd 7060 df-sitm 29720 |
This theorem is referenced by: sitmcl 29740 sitmf 29741 |
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