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Theorem sitmfval 29545
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 29544 . 2 (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔))))
5 simprl 789 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑓 = 𝐹)
6 simprr 791 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑔 = 𝐺)
75, 6oveq12d 6545 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑓𝑓 𝐷𝑔) = (𝐹𝑓 𝐷𝐺))
87fveq2d 6092 . 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 6098 . . 3 (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 𝐷𝐺)) ∈ V
1211a1i 11 . 2 (𝜑 → (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 𝐷𝐺)) ∈ V)
134, 8, 9, 10, 12ovmpt2d 6664 1 (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹𝑓 𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172   cuni 4366  dom cdm 5028  ran crn 5029  cfv 5790  (class class class)co 6527  𝑓 cof 6770  0cc0 9792  +∞cpnf 9927  [,]cicc 12005  s cress 15642  distcds 15723  *𝑠cxrs 15929  measurescmeas 29391  sitmcsitm 29523  sitgcsitg 29524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-1st 7036  df-2nd 7037  df-sitm 29526
This theorem is referenced by:  sitmcl  29546  sitmf  29547
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