Theorem sitmval 34493

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 3450	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑊 ∈ V)
4		sitmval.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
5		oveq1 7374	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
6	5	dmeqd 5860	. . . 4 ⊢ (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7		fveq2 6840	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8	7	ofeqd 7633	. . . . . 6 ⊢ (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
9	8	oveqd 7384	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (dist‘𝑤)𝑔) = (𝑓 ∘f (dist‘𝑊)𝑔))
10	9	fveq2d 6844	. . . 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 7442	. . 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 7375	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
13	12	dmeqd 5860	. . . 4 ⊢ (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
14		oveq2 7375	. . . . 5 ⊢ (𝑚 = 𝑀 → ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀))
15		sitmval.d	. . . . . . . 8 ⊢ 𝐷 = (dist‘𝑊)
16	15	eqcomi 2745	. . . . . . 7 ⊢ (dist‘𝑊) = 𝐷
17		ofeq 7634	. . . . . . 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 7384	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑓 ∘f (dist‘𝑊)𝑔) = (𝑓 ∘f 𝐷𝑔))
20	14, 19	fveq12d 6847	. . . 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 7442	. . 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 34475	. . 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 7400	. . . . 5 ⊢ (𝑊sitg𝑀) ∈ V
24	23	dmex 7860	. . . 4 ⊢ dom (𝑊sitg𝑀) ∈ V
25	24, 24	mpoex 8032	. . 3 ⊢ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))) ∈ V
26	11, 21, 22, 25	ovmpo 7527	. 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 585	1 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∪ cuni 4850  dom cdm 5631  ran crn 5632  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ∘_f cof 7629  0cc0 11038  +∞cpnf 11176  [,]cicc 13301   ↾_s cress 17200  distcds 17229  ℝ^*_𝑠cxrs 17464  measurescmeas 34339  sitmcsitm 34472  sitgcsitg 34473

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-1st 7942  df-2nd 7943  df-sitm 34475

This theorem is referenced by:  sitmfval  34494  sitmf  34496