Theorem sitmval 33336

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	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitmval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)

Assertion

Ref	Expression
sitmval	⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 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.

Step	Hyp	Ref	Expression
1		sitmval.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
2		elex 3492	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑊 ∈ V)
4		sitmval.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
5		oveq1 7412	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
6	5	dmeqd 5903	. . . 4 ⊢ (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7		fveq2 6888	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8	7	ofeqd 7668	. . . . . 6 ⊢ (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
9	8	oveqd 7422	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (dist‘𝑤)𝑔) = (𝑓 ∘f (dist‘𝑊)𝑔))
10	9	fveq2d 6892	. . . 4 ⊢ (𝑤 = 𝑊 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔)) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑊)𝑔)))
11	6, 6, 10	mpoeq123dv 7480	. . 3 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔))) = (𝑓 ∈ dom (𝑊sitg𝑚), 𝑔 ∈ dom (𝑊sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑊)𝑔))))
12		oveq2 7413	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
13	12	dmeqd 5903	. . . 4 ⊢ (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
14		oveq2 7413	. . . . 5 ⊢ (𝑚 = 𝑀 → ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀))
15		sitmval.d	. . . . . . . 8 ⊢ 𝐷 = (dist‘𝑊)
16	15	eqcomi 2741	. . . . . . 7 ⊢ (dist‘𝑊) = 𝐷
17		ofeq 7669	. . . . . . 7 ⊢ ((dist‘𝑊) = 𝐷 → ∘f (dist‘𝑊) = ∘f 𝐷)
18	16, 17	mp1i 13	. . . . . 6 ⊢ (𝑚 = 𝑀 → ∘f (dist‘𝑊) = ∘f 𝐷)
19	18	oveqd 7422	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑓 ∘f (dist‘𝑊)𝑔) = (𝑓 ∘f 𝐷𝑔))
20	14, 19	fveq12d 6895	. . . 4 ⊢ (𝑚 = 𝑀 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑊)𝑔)) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔)))
21	13, 13, 20	mpoeq123dv 7480	. . 3 ⊢ (𝑚 = 𝑀 → (𝑓 ∈ dom (𝑊sitg𝑚), 𝑔 ∈ dom (𝑊sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑊)𝑔))) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))
22		df-sitm 33318	. . 3 ⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔))))
23		ovex 7438	. . . . 5 ⊢ (𝑊sitg𝑀) ∈ V
24	23	dmex 7898	. . . 4 ⊢ dom (𝑊sitg𝑀) ∈ V
25	24, 24	mpoex 8062	. . 3 ⊢ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))) ∈ V
26	11, 21, 22, 25	ovmpo 7564	. 2 ⊢ ((𝑊 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))
27	3, 4, 26	syl2anc 584	1 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∪ cuni 4907  dom cdm 5675  ran crn 5676  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ∘_f cof 7664  0cc0 11106  +∞cpnf 11241  [,]cicc 13323   ↾_s cress 17169  distcds 17202  ℝ^*_𝑠cxrs 17442  measurescmeas 33181  sitmcsitm 33315  sitgcsitg 33316

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-1st 7971  df-2nd 7972  df-sitm 33318

This theorem is referenced by:  sitmfval  33337  sitmf  33339