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Theorem sitmval 32761
Description: Value of the simple function integral metric for a given space π‘Š and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
sitmval.d 𝐷 = (distβ€˜π‘Š)
sitmval.1 (πœ‘ β†’ π‘Š ∈ 𝑉)
sitmval.2 (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)
Assertion
Ref Expression
sitmval (πœ‘ β†’ (π‘Šsitm𝑀) = (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))))
Distinct variable groups:   𝑓,𝑔,𝑀   𝑓,π‘Š,𝑔
Allowed substitution hints:   πœ‘(𝑓,𝑔)   𝐷(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem sitmval
Dummy variables 𝑀 π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitmval.1 . . 3 (πœ‘ β†’ π‘Š ∈ 𝑉)
2 elex 3461 . . 3 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
31, 2syl 17 . 2 (πœ‘ β†’ π‘Š ∈ V)
4 sitmval.2 . 2 (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)
5 oveq1 7358 . . . . 5 (𝑀 = π‘Š β†’ (𝑀sitgπ‘š) = (π‘Šsitgπ‘š))
65dmeqd 5859 . . . 4 (𝑀 = π‘Š β†’ dom (𝑀sitgπ‘š) = dom (π‘Šsitgπ‘š))
7 fveq2 6839 . . . . . . 7 (𝑀 = π‘Š β†’ (distβ€˜π‘€) = (distβ€˜π‘Š))
87ofeqd 7611 . . . . . 6 (𝑀 = π‘Š β†’ ∘f (distβ€˜π‘€) = ∘f (distβ€˜π‘Š))
98oveqd 7368 . . . . 5 (𝑀 = π‘Š β†’ (𝑓 ∘f (distβ€˜π‘€)𝑔) = (𝑓 ∘f (distβ€˜π‘Š)𝑔))
109fveq2d 6843 . . . 4 (𝑀 = π‘Š β†’ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š)β€˜(𝑓 ∘f (distβ€˜π‘€)𝑔)) = (((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š)β€˜(𝑓 ∘f (distβ€˜π‘Š)𝑔)))
116, 6, 10mpoeq123dv 7426 . . 3 (𝑀 = π‘Š β†’ (𝑓 ∈ dom (𝑀sitgπ‘š), 𝑔 ∈ dom (𝑀sitgπ‘š) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š)β€˜(𝑓 ∘f (distβ€˜π‘€)𝑔))) = (𝑓 ∈ dom (π‘Šsitgπ‘š), 𝑔 ∈ dom (π‘Šsitgπ‘š) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š)β€˜(𝑓 ∘f (distβ€˜π‘Š)𝑔))))
12 oveq2 7359 . . . . 5 (π‘š = 𝑀 β†’ (π‘Šsitgπ‘š) = (π‘Šsitg𝑀))
1312dmeqd 5859 . . . 4 (π‘š = 𝑀 β†’ dom (π‘Šsitgπ‘š) = dom (π‘Šsitg𝑀))
14 oveq2 7359 . . . . 5 (π‘š = 𝑀 β†’ ((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š) = ((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀))
15 sitmval.d . . . . . . . 8 𝐷 = (distβ€˜π‘Š)
1615eqcomi 2746 . . . . . . 7 (distβ€˜π‘Š) = 𝐷
17 ofeq 7612 . . . . . . 7 ((distβ€˜π‘Š) = 𝐷 β†’ ∘f (distβ€˜π‘Š) = ∘f 𝐷)
1816, 17mp1i 13 . . . . . 6 (π‘š = 𝑀 β†’ ∘f (distβ€˜π‘Š) = ∘f 𝐷)
1918oveqd 7368 . . . . 5 (π‘š = 𝑀 β†’ (𝑓 ∘f (distβ€˜π‘Š)𝑔) = (𝑓 ∘f 𝐷𝑔))
2014, 19fveq12d 6846 . . . 4 (π‘š = 𝑀 β†’ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š)β€˜(𝑓 ∘f (distβ€˜π‘Š)𝑔)) = (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔)))
2113, 13, 20mpoeq123dv 7426 . . 3 (π‘š = 𝑀 β†’ (𝑓 ∈ dom (π‘Šsitgπ‘š), 𝑔 ∈ dom (π‘Šsitgπ‘š) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š)β€˜(𝑓 ∘f (distβ€˜π‘Š)𝑔))) = (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))))
22 df-sitm 32743 . . 3 sitm = (𝑀 ∈ V, π‘š ∈ βˆͺ ran measures ↦ (𝑓 ∈ dom (𝑀sitgπ‘š), 𝑔 ∈ dom (𝑀sitgπ‘š) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitgπ‘š)β€˜(𝑓 ∘f (distβ€˜π‘€)𝑔))))
23 ovex 7384 . . . . 5 (π‘Šsitg𝑀) ∈ V
2423dmex 7840 . . . 4 dom (π‘Šsitg𝑀) ∈ V
2524, 24mpoex 8004 . . 3 (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))) ∈ V
2611, 21, 22, 25ovmpo 7509 . 2 ((π‘Š ∈ V ∧ 𝑀 ∈ βˆͺ ran measures) β†’ (π‘Šsitm𝑀) = (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))))
273, 4, 26syl2anc 584 1 (πœ‘ β†’ (π‘Šsitm𝑀) = (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3443  βˆͺ cuni 4863  dom cdm 5631  ran crn 5632  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353   ∘f cof 7607  0cc0 11009  +∞cpnf 11144  [,]cicc 13221   β†Ύs cress 17072  distcds 17102  β„*𝑠cxrs 17342  measurescmeas 32606  sitmcsitm 32740  sitgcsitg 32741
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609  df-1st 7913  df-2nd 7914  df-sitm 32743
This theorem is referenced by:  sitmfval  32762  sitmf  32764
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