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Theorem sitmval 31224
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 3455 . . 3 (𝑊𝑉𝑊 ∈ V)
31, 2syl 17 . 2 (𝜑𝑊 ∈ V)
4 sitmval.2 . 2 (𝜑𝑀 ran measures)
5 oveq1 7023 . . . . 5 (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
65dmeqd 5660 . . . 4 (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7 fveq2 6538 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8 ofeq 7269 . . . . . . 7 ((dist‘𝑤) = (dist‘𝑊) → ∘𝑓 (dist‘𝑤) = ∘𝑓 (dist‘𝑊))
97, 8syl 17 . . . . . 6 (𝑤 = 𝑊 → ∘𝑓 (dist‘𝑤) = ∘𝑓 (dist‘𝑊))
109oveqd 7033 . . . . 5 (𝑤 = 𝑊 → (𝑓𝑓 (dist‘𝑤)𝑔) = (𝑓𝑓 (dist‘𝑊)𝑔))
1110fveq2d 6542 . . . 4 (𝑤 = 𝑊 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑊)𝑔)))
126, 6, 11mpoeq123dv 7087 . . 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 7024 . . . . 5 (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
1413dmeqd 5660 . . . 4 (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
15 oveq2 7024 . . . . 5 (𝑚 = 𝑀 → ((ℝ*𝑠s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠s (0[,]+∞))sitg𝑀))
16 sitmval.d . . . . . . . 8 𝐷 = (dist‘𝑊)
1716eqcomi 2804 . . . . . . 7 (dist‘𝑊) = 𝐷
18 ofeq 7269 . . . . . . 7 ((dist‘𝑊) = 𝐷 → ∘𝑓 (dist‘𝑊) = ∘𝑓 𝐷)
1917, 18mp1i 13 . . . . . 6 (𝑚 = 𝑀 → ∘𝑓 (dist‘𝑊) = ∘𝑓 𝐷)
2019oveqd 7033 . . . . 5 (𝑚 = 𝑀 → (𝑓𝑓 (dist‘𝑊)𝑔) = (𝑓𝑓 𝐷𝑔))
2115, 20fveq12d 6545 . . . 4 (𝑚 = 𝑀 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑊)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔)))
2214, 14, 21mpoeq123dv 7087 . . 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 31206 . . 3 sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔))))
24 ovex 7048 . . . . 5 (𝑊sitg𝑀) ∈ V
2524dmex 7472 . . . 4 dom (𝑊sitg𝑀) ∈ V
2625, 25mpoex 7633 . . 3 (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔))) ∈ V
2712, 22, 23, 26ovmpo 7166 . 2 ((𝑊 ∈ V ∧ 𝑀 ran measures) → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔))))
283, 4, 27syl2anc 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 1522  wcel 2081  Vcvv 3437   cuni 4745  dom cdm 5443  ran crn 5444  cfv 6225  (class class class)co 7016  cmpo 7018  𝑓 cof 7265  0cc0 10383  +∞cpnf 10518  [,]cicc 12591  s cress 16313  distcds 16403  *𝑠cxrs 16602  measurescmeas 31071  sitmcsitm 31203  sitgcsitg 31204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-1st 7545  df-2nd 7546  df-sitm 31206
This theorem is referenced by:  sitmfval  31225  sitmf  31227
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