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Theorem sitmfval 32990
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 32989 . 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 770 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ 𝑓 = 𝐹)
6 simprr 772 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ 𝑔 = 𝐺)
75, 6oveq12d 7380 . . 3 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ (𝑓 ∘f 𝐷𝑔) = (𝐹 ∘f 𝐷𝐺))
87fveq2d 6851 . 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 6862 . 2 (πœ‘ β†’ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝐹 ∘f 𝐷𝐺)) ∈ V)
124, 8, 9, 10, 11ovmpod 7512 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3448  βˆͺ cuni 4870  dom cdm 5638  ran crn 5639  β€˜cfv 6501  (class class class)co 7362   ∘f cof 7620  0cc0 11058  +∞cpnf 11193  [,]cicc 13274   β†Ύs cress 17119  distcds 17149  β„*𝑠cxrs 17389  measurescmeas 32834  sitmcsitm 32968  sitgcsitg 32969
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-1st 7926  df-2nd 7927  df-sitm 32971
This theorem is referenced by:  sitmcl  32991  sitmf  32992
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