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Theorem sitmval 29539
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 3179 . . 3 (𝑊𝑉𝑊 ∈ V)
31, 2syl 17 . 2 (𝜑𝑊 ∈ V)
4 sitmval.2 . 2 (𝜑𝑀 ran measures)
5 oveq1 6529 . . . . 5 (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
65dmeqd 5230 . . . 4 (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7 fveq2 6083 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8 ofeq 6769 . . . . . . 7 ((dist‘𝑤) = (dist‘𝑊) → ∘𝑓 (dist‘𝑤) = ∘𝑓 (dist‘𝑊))
97, 8syl 17 . . . . . 6 (𝑤 = 𝑊 → ∘𝑓 (dist‘𝑤) = ∘𝑓 (dist‘𝑊))
109oveqd 6539 . . . . 5 (𝑤 = 𝑊 → (𝑓𝑓 (dist‘𝑤)𝑔) = (𝑓𝑓 (dist‘𝑊)𝑔))
1110fveq2d 6087 . . . 4 (𝑤 = 𝑊 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑊)𝑔)))
126, 6, 11mpt2eq123dv 6588 . . 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 6530 . . . . 5 (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
1413dmeqd 5230 . . . 4 (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
15 oveq2 6530 . . . . 5 (𝑚 = 𝑀 → ((ℝ*𝑠s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠s (0[,]+∞))sitg𝑀))
16 sitmval.d . . . . . . . 8 𝐷 = (dist‘𝑊)
1716eqcomi 2613 . . . . . . 7 (dist‘𝑊) = 𝐷
18 ofeq 6769 . . . . . . 7 ((dist‘𝑊) = 𝐷 → ∘𝑓 (dist‘𝑊) = ∘𝑓 𝐷)
1917, 18mp1i 13 . . . . . 6 (𝑚 = 𝑀 → ∘𝑓 (dist‘𝑊) = ∘𝑓 𝐷)
2019oveqd 6539 . . . . 5 (𝑚 = 𝑀 → (𝑓𝑓 (dist‘𝑊)𝑔) = (𝑓𝑓 𝐷𝑔))
2115, 20fveq12d 6089 . . . 4 (𝑚 = 𝑀 → (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑊)𝑔)) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔)))
2214, 14, 21mpt2eq123dv 6588 . . 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 29521 . . 3 sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔))))
24 ovex 6550 . . . . 5 (𝑊sitg𝑀) ∈ V
2524dmex 6963 . . . 4 dom (𝑊sitg𝑀) ∈ V
2625, 25mpt2ex 7108 . . 3 (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔))) ∈ V
2712, 22, 23, 26ovmpt2 6667 . 2 ((𝑊 ∈ V ∧ 𝑀 ran measures) → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓𝑓 𝐷𝑔))))
283, 4, 27syl2anc 690 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 1474  wcel 1975  Vcvv 3167   cuni 4361  dom cdm 5023  ran crn 5024  cfv 5785  (class class class)co 6522  cmpt2 6524  𝑓 cof 6765  0cc0 9787  +∞cpnf 9922  [,]cicc 12000  s cress 15637  distcds 15718  *𝑠cxrs 15924  measurescmeas 29386  sitmcsitm 29518  sitgcsitg 29519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-1st 7031  df-2nd 7032  df-sitm 29521
This theorem is referenced by:  sitmfval  29540  sitmf  29542
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