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Theorem sitmval 33803
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 3485 . . 3 (𝑊𝑉𝑊 ∈ V)
31, 2syl 17 . 2 (𝜑𝑊 ∈ V)
4 sitmval.2 . 2 (𝜑𝑀 ran measures)
5 oveq1 7408 . . . . 5 (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
65dmeqd 5895 . . . 4 (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7 fveq2 6881 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
87ofeqd 7665 . . . . . 6 (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
98oveqd 7418 . . . . 5 (𝑤 = 𝑊 → (𝑓f (dist‘𝑤)𝑔) = (𝑓f (dist‘𝑊)𝑔))
109fveq2d 6885 . . . 4 (𝑤 = 𝑊 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔)))
116, 6, 10mpoeq123dv 7476 . . 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 7409 . . . . 5 (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
1312dmeqd 5895 . . . 4 (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
14 oveq2 7409 . . . . 5 (𝑚 = 𝑀 → ((ℝ*𝑠s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠s (0[,]+∞))sitg𝑀))
15 sitmval.d . . . . . . . 8 𝐷 = (dist‘𝑊)
1615eqcomi 2733 . . . . . . 7 (dist‘𝑊) = 𝐷
17 ofeq 7666 . . . . . . 7 ((dist‘𝑊) = 𝐷 → ∘f (dist‘𝑊) = ∘f 𝐷)
1816, 17mp1i 13 . . . . . 6 (𝑚 = 𝑀 → ∘f (dist‘𝑊) = ∘f 𝐷)
1918oveqd 7418 . . . . 5 (𝑚 = 𝑀 → (𝑓f (dist‘𝑊)𝑔) = (𝑓f 𝐷𝑔))
2014, 19fveq12d 6888 . . . 4 (𝑚 = 𝑀 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔)))
2113, 13, 20mpoeq123dv 7476 . . 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 33785 . . 3 sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔))))
23 ovex 7434 . . . . 5 (𝑊sitg𝑀) ∈ V
2423dmex 7895 . . . 4 dom (𝑊sitg𝑀) ∈ V
2524, 24mpoex 8059 . . 3 (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))) ∈ V
2611, 21, 22, 25ovmpo 7560 . 2 ((𝑊 ∈ V ∧ 𝑀 ran measures) → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
273, 4, 26syl2anc 583 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 1533  wcel 2098  Vcvv 3466   cuni 4899  dom cdm 5666  ran crn 5667  cfv 6533  (class class class)co 7401  cmpo 7403  f cof 7661  0cc0 11105  +∞cpnf 11241  [,]cicc 13323  s cress 17169  distcds 17202  *𝑠cxrs 17442  measurescmeas 33648  sitmcsitm 33782  sitgcsitg 33783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-1st 7968  df-2nd 7969  df-sitm 33785
This theorem is referenced by:  sitmfval  33804  sitmf  33806
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