Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sitmval Structured version   Visualization version   GIF version

Theorem sitmval 34351
Description: Value of the simple function integral metric for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
sitmval.d 𝐷 = (dist‘𝑊)
sitmval.1 (𝜑𝑊𝑉)
sitmval.2 (𝜑𝑀 ran measures)
Assertion
Ref Expression
sitmval (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
Distinct variable groups:   𝑓,𝑔,𝑀   𝑓,𝑊,𝑔
Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐷(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem sitmval
Dummy variables 𝑤 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitmval.1 . . 3 (𝜑𝑊𝑉)
2 elex 3501 . . 3 (𝑊𝑉𝑊 ∈ V)
31, 2syl 17 . 2 (𝜑𝑊 ∈ V)
4 sitmval.2 . 2 (𝜑𝑀 ran measures)
5 oveq1 7438 . . . . 5 (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
65dmeqd 5916 . . . 4 (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7 fveq2 6906 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
87ofeqd 7699 . . . . . 6 (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
98oveqd 7448 . . . . 5 (𝑤 = 𝑊 → (𝑓f (dist‘𝑤)𝑔) = (𝑓f (dist‘𝑊)𝑔))
109fveq2d 6910 . . . 4 (𝑤 = 𝑊 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔)))
116, 6, 10mpoeq123dv 7508 . . 3 (𝑤 = 𝑊 → (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔))) = (𝑓 ∈ dom (𝑊sitg𝑚), 𝑔 ∈ dom (𝑊sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔))))
12 oveq2 7439 . . . . 5 (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
1312dmeqd 5916 . . . 4 (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
14 oveq2 7439 . . . . 5 (𝑚 = 𝑀 → ((ℝ*𝑠s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠s (0[,]+∞))sitg𝑀))
15 sitmval.d . . . . . . . 8 𝐷 = (dist‘𝑊)
1615eqcomi 2746 . . . . . . 7 (dist‘𝑊) = 𝐷
17 ofeq 7700 . . . . . . 7 ((dist‘𝑊) = 𝐷 → ∘f (dist‘𝑊) = ∘f 𝐷)
1816, 17mp1i 13 . . . . . 6 (𝑚 = 𝑀 → ∘f (dist‘𝑊) = ∘f 𝐷)
1918oveqd 7448 . . . . 5 (𝑚 = 𝑀 → (𝑓f (dist‘𝑊)𝑔) = (𝑓f 𝐷𝑔))
2014, 19fveq12d 6913 . . . 4 (𝑚 = 𝑀 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔)))
2113, 13, 20mpoeq123dv 7508 . . 3 (𝑚 = 𝑀 → (𝑓 ∈ dom (𝑊sitg𝑚), 𝑔 ∈ dom (𝑊sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔))) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
22 df-sitm 34333 . . 3 sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔))))
23 ovex 7464 . . . . 5 (𝑊sitg𝑀) ∈ V
2423dmex 7931 . . . 4 dom (𝑊sitg𝑀) ∈ V
2524, 24mpoex 8104 . . 3 (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))) ∈ V
2611, 21, 22, 25ovmpo 7593 . 2 ((𝑊 ∈ V ∧ 𝑀 ran measures) → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
273, 4, 26syl2anc 584 1 (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480   cuni 4907  dom cdm 5685  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  f cof 7695  0cc0 11155  +∞cpnf 11292  [,]cicc 13390  s cress 17274  distcds 17306  *𝑠cxrs 17545  measurescmeas 34196  sitmcsitm 34330  sitgcsitg 34331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8014  df-2nd 8015  df-sitm 34333
This theorem is referenced by:  sitmfval  34352  sitmf  34354
  Copyright terms: Public domain W3C validator