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Theorem sitmval 33646
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 3491 . . 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 7418 . . . . 5 (𝑀 = π‘Š β†’ (𝑀sitgπ‘š) = (π‘Šsitgπ‘š))
65dmeqd 5904 . . . 4 (𝑀 = π‘Š β†’ dom (𝑀sitgπ‘š) = dom (π‘Šsitgπ‘š))
7 fveq2 6890 . . . . . . 7 (𝑀 = π‘Š β†’ (distβ€˜π‘€) = (distβ€˜π‘Š))
87ofeqd 7674 . . . . . 6 (𝑀 = π‘Š β†’ ∘f (distβ€˜π‘€) = ∘f (distβ€˜π‘Š))
98oveqd 7428 . . . . 5 (𝑀 = π‘Š β†’ (𝑓 ∘f (distβ€˜π‘€)𝑔) = (𝑓 ∘f (distβ€˜π‘Š)𝑔))
109fveq2d 6894 . . . 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 7486 . . 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 7419 . . . . 5 (π‘š = 𝑀 β†’ (π‘Šsitgπ‘š) = (π‘Šsitg𝑀))
1312dmeqd 5904 . . . 4 (π‘š = 𝑀 β†’ dom (π‘Šsitgπ‘š) = dom (π‘Šsitg𝑀))
14 oveq2 7419 . . . . 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 2739 . . . . . . 7 (distβ€˜π‘Š) = 𝐷
17 ofeq 7675 . . . . . . 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 7428 . . . . 5 (π‘š = 𝑀 β†’ (𝑓 ∘f (distβ€˜π‘Š)𝑔) = (𝑓 ∘f 𝐷𝑔))
2014, 19fveq12d 6897 . . . 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 7486 . . 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 33628 . . 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 7444 . . . . 5 (π‘Šsitg𝑀) ∈ V
2423dmex 7904 . . . 4 dom (π‘Šsitg𝑀) ∈ V
2524, 24mpoex 8068 . . 3 (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))) ∈ V
2611, 21, 22, 25ovmpo 7570 . 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 582 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 1539   ∈ wcel 2104  Vcvv 3472  βˆͺ cuni 4907  dom cdm 5675  ran crn 5676  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413   ∘f cof 7670  0cc0 11112  +∞cpnf 11249  [,]cicc 13331   β†Ύs cress 17177  distcds 17210  β„*𝑠cxrs 17450  measurescmeas 33491  sitmcsitm 33625  sitgcsitg 33626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-1st 7977  df-2nd 7978  df-sitm 33628
This theorem is referenced by:  sitmfval  33647  sitmf  33649
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