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Theorem sitmval 31847
 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 3428 . . 3 (𝑊𝑉𝑊 ∈ V)
31, 2syl 17 . 2 (𝜑𝑊 ∈ V)
4 sitmval.2 . 2 (𝜑𝑀 ran measures)
5 oveq1 7163 . . . . 5 (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
65dmeqd 5751 . . . 4 (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7 fveq2 6663 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8 ofeq 7413 . . . . . . 7 ((dist‘𝑤) = (dist‘𝑊) → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
97, 8syl 17 . . . . . 6 (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
109oveqd 7173 . . . . 5 (𝑤 = 𝑊 → (𝑓f (dist‘𝑤)𝑔) = (𝑓f (dist‘𝑊)𝑔))
1110fveq2d 6667 . . . 4 (𝑤 = 𝑊 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔)))
126, 6, 11mpoeq123dv 7229 . . 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))))
13 oveq2 7164 . . . . 5 (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
1413dmeqd 5751 . . . 4 (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
15 oveq2 7164 . . . . 5 (𝑚 = 𝑀 → ((ℝ*𝑠s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠s (0[,]+∞))sitg𝑀))
16 sitmval.d . . . . . . . 8 𝐷 = (dist‘𝑊)
1716eqcomi 2767 . . . . . . 7 (dist‘𝑊) = 𝐷
18 ofeq 7413 . . . . . . 7 ((dist‘𝑊) = 𝐷 → ∘f (dist‘𝑊) = ∘f 𝐷)
1917, 18mp1i 13 . . . . . 6 (𝑚 = 𝑀 → ∘f (dist‘𝑊) = ∘f 𝐷)
2019oveqd 7173 . . . . 5 (𝑚 = 𝑀 → (𝑓f (dist‘𝑊)𝑔) = (𝑓f 𝐷𝑔))
2115, 20fveq12d 6670 . . . 4 (𝑚 = 𝑀 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑊)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔)))
2214, 14, 21mpoeq123dv 7229 . . 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))))
23 df-sitm 31829 . . 3 sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔))))
24 ovex 7189 . . . . 5 (𝑊sitg𝑀) ∈ V
2524dmex 7627 . . . 4 dom (𝑊sitg𝑀) ∈ V
2625, 25mpoex 7788 . . 3 (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))) ∈ V
2712, 22, 23, 26ovmpo 7311 . 2 ((𝑊 ∈ V ∧ 𝑀 ran measures) → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
283, 4, 27syl2anc 587 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 1538   ∈ wcel 2111  Vcvv 3409  ∪ cuni 4801  dom cdm 5528  ran crn 5529  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158   ∘f cof 7409  0cc0 10588  +∞cpnf 10723  [,]cicc 12795   ↾s cress 16555  distcds 16645  ℝ*𝑠cxrs 16844  measurescmeas 31694  sitmcsitm 31826  sitgcsitg 31827 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-1st 7699  df-2nd 7700  df-sitm 31829 This theorem is referenced by:  sitmfval  31848  sitmf  31850
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