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Theorem sitmval 33348
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 3493 . . 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 7416 . . . . 5 (𝑀 = π‘Š β†’ (𝑀sitgπ‘š) = (π‘Šsitgπ‘š))
65dmeqd 5906 . . . 4 (𝑀 = π‘Š β†’ dom (𝑀sitgπ‘š) = dom (π‘Šsitgπ‘š))
7 fveq2 6892 . . . . . . 7 (𝑀 = π‘Š β†’ (distβ€˜π‘€) = (distβ€˜π‘Š))
87ofeqd 7672 . . . . . 6 (𝑀 = π‘Š β†’ ∘f (distβ€˜π‘€) = ∘f (distβ€˜π‘Š))
98oveqd 7426 . . . . 5 (𝑀 = π‘Š β†’ (𝑓 ∘f (distβ€˜π‘€)𝑔) = (𝑓 ∘f (distβ€˜π‘Š)𝑔))
109fveq2d 6896 . . . 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 7484 . . 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 7417 . . . . 5 (π‘š = 𝑀 β†’ (π‘Šsitgπ‘š) = (π‘Šsitg𝑀))
1312dmeqd 5906 . . . 4 (π‘š = 𝑀 β†’ dom (π‘Šsitgπ‘š) = dom (π‘Šsitg𝑀))
14 oveq2 7417 . . . . 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 2742 . . . . . . 7 (distβ€˜π‘Š) = 𝐷
17 ofeq 7673 . . . . . . 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 7426 . . . . 5 (π‘š = 𝑀 β†’ (𝑓 ∘f (distβ€˜π‘Š)𝑔) = (𝑓 ∘f 𝐷𝑔))
2014, 19fveq12d 6899 . . . 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 7484 . . 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 33330 . . 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 7442 . . . . 5 (π‘Šsitg𝑀) ∈ V
2423dmex 7902 . . . 4 dom (π‘Šsitg𝑀) ∈ V
2524, 24mpoex 8066 . . 3 (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))) ∈ V
2611, 21, 22, 25ovmpo 7568 . 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 585 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 1542   ∈ wcel 2107  Vcvv 3475  βˆͺ cuni 4909  dom cdm 5677  ran crn 5678  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   ∘f cof 7668  0cc0 11110  +∞cpnf 11245  [,]cicc 13327   β†Ύs cress 17173  distcds 17206  β„*𝑠cxrs 17446  measurescmeas 33193  sitmcsitm 33327  sitgcsitg 33328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-1st 7975  df-2nd 7976  df-sitm 33330
This theorem is referenced by:  sitmfval  33349  sitmf  33351
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