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Theorem sitmfval 33338
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 33337 . 2 (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
5 simprl 770 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑓 = 𝐹)
6 simprr 772 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑔 = 𝐺)
75, 6oveq12d 7424 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑓f 𝐷𝑔) = (𝐹f 𝐷𝐺))
87fveq2d 6893 . 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 fvexd 6904 . 2 (𝜑 → (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹f 𝐷𝐺)) ∈ V)
124, 8, 9, 10, 11ovmpod 7557 1 (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹f 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  *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
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