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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 30539 | . 2 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘𝑓 𝐷𝑔)))) |
5 | simprl 809 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹) | |
6 | simprr 811 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺) | |
7 | 5, 6 | oveq12d 6708 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 ∘𝑓 𝐷𝑔) = (𝐹 ∘𝑓 𝐷𝐺)) |
8 | 7 | fveq2d 6233 | . 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 6241 | . 2 ⊢ (𝜑 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)) ∈ V) | |
12 | 4, 8, 9, 10, 11 | ovmpt2d 6830 | 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 1523 ∈ wcel 2030 Vcvv 3231 ∪ cuni 4468 dom cdm 5143 ran crn 5144 ‘cfv 5926 (class class class)co 6690 ∘𝑓 cof 6937 0cc0 9974 +∞cpnf 10109 [,]cicc 12216 ↾s cress 15905 distcds 15997 ℝ*𝑠cxrs 16207 measurescmeas 30386 sitmcsitm 30518 sitgcsitg 30519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-1st 7210 df-2nd 7211 df-sitm 30521 |
This theorem is referenced by: sitmcl 30541 sitmf 30542 |
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