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Theorem sitmfval 29739
 Description: Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
sitmval.d 𝐷 = (dist‘𝑊)
sitmval.1 (𝜑𝑊𝑉)
sitmval.2 (𝜑𝑀 ran measures)
sitmfval.1 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
sitmfval.2 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
Assertion
Ref Expression
sitmfval (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 𝐷𝐺)))

Proof of Theorem sitmfval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitmval.d . . 3 𝐷 = (dist‘𝑊)
2 sitmval.1 . . 3 (𝜑𝑊𝑉)
3 sitmval.2 . . 3 (𝜑𝑀 ran measures)
41, 2, 3sitmval 29738 . 2 (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔))))
5 simprl 790 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑓 = 𝐹)
6 simprr 792 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑔 = 𝐺)
75, 6oveq12d 6567 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑓𝑓 𝐷𝑔) = (𝐹𝑓 𝐷𝐺))
87fveq2d 6107 . 2 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 𝐷𝐺)))
9 sitmfval.1 . 2 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
10 sitmfval.2 . 2 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
11 fvex 6113 . . 3 (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 𝐷𝐺)) ∈ V
1211a1i 11 . 2 (𝜑 → (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 𝐷𝐺)) ∈ V)
134, 8, 9, 10, 12ovmpt2d 6686 1 (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 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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