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Theorem sitmfval 33648
Description: Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
sitmval.d 𝐷 = (distβ€˜π‘Š)
sitmval.1 (πœ‘ β†’ π‘Š ∈ 𝑉)
sitmval.2 (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)
sitmfval.1 (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))
sitmfval.2 (πœ‘ β†’ 𝐺 ∈ dom (π‘Šsitg𝑀))
Assertion
Ref Expression
sitmfval (πœ‘ β†’ (𝐹(π‘Šsitm𝑀)𝐺) = (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝐹 ∘f 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 = (distβ€˜π‘Š)
2 sitmval.1 . . 3 (πœ‘ β†’ π‘Š ∈ 𝑉)
3 sitmval.2 . . 3 (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)
41, 2, 3sitmval 33647 . 2 (πœ‘ β†’ (π‘Šsitm𝑀) = (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))))
5 simprl 768 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ 𝑓 = 𝐹)
6 simprr 770 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ 𝑔 = 𝐺)
75, 6oveq12d 7430 . . 3 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ (𝑓 ∘f 𝐷𝑔) = (𝐹 ∘f 𝐷𝐺))
87fveq2d 6895 . 2 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔)) = (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝐹 ∘f 𝐷𝐺)))
9 sitmfval.1 . 2 (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))
10 sitmfval.2 . 2 (πœ‘ β†’ 𝐺 ∈ dom (π‘Šsitg𝑀))
11 fvexd 6906 . 2 (πœ‘ β†’ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝐹 ∘f 𝐷𝐺)) ∈ V)
124, 8, 9, 10, 11ovmpod 7563 1 (πœ‘ β†’ (𝐹(π‘Šsitm𝑀)𝐺) = (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝐹 ∘f 𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473  βˆͺ cuni 4908  dom cdm 5676  ran crn 5677  β€˜cfv 6543  (class class class)co 7412   ∘f cof 7671  0cc0 11113  +∞cpnf 11250  [,]cicc 13332   β†Ύs cress 17178  distcds 17211  β„*𝑠cxrs 17451  measurescmeas 33492  sitmcsitm 33626  sitgcsitg 33627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-1st 7978  df-2nd 7979  df-sitm 33629
This theorem is referenced by:  sitmcl  33649  sitmf  33650
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