Theorem sitmval 32216

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 3440	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑊 ∈ V)
4		sitmval.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
5		oveq1 7262	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
6	5	dmeqd 5803	. . . 4 ⊢ (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7		fveq2 6756	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8	7	ofeqd 7513	. . . . . 6 ⊢ (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
9	8	oveqd 7272	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (dist‘𝑤)𝑔) = (𝑓 ∘f (dist‘𝑊)𝑔))
10	9	fveq2d 6760	. . . 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 7328	. . 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 7263	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
13	12	dmeqd 5803	. . . 4 ⊢ (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
14		oveq2 7263	. . . . 5 ⊢ (𝑚 = 𝑀 → ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀))
15		sitmval.d	. . . . . . . 8 ⊢ 𝐷 = (dist‘𝑊)
16	15	eqcomi 2747	. . . . . . 7 ⊢ (dist‘𝑊) = 𝐷
17		ofeq 7514	. . . . . . 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 7272	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑓 ∘f (dist‘𝑊)𝑔) = (𝑓 ∘f 𝐷𝑔))
20	14, 19	fveq12d 6763	. . . 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 7328	. . 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 32198	. . 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 7288	. . . . 5 ⊢ (𝑊sitg𝑀) ∈ V
24	23	dmex 7732	. . . 4 ⊢ dom (𝑊sitg𝑀) ∈ V
25	24, 24	mpoex 7893	. . 3 ⊢ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))) ∈ V
26	11, 21, 22, 25	ovmpo 7411	. 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 583	1 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∪ cuni 4836  dom cdm 5580  ran crn 5581  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ∘_f cof 7509  0cc0 10802  +∞cpnf 10937  [,]cicc 13011   ↾_s cress 16867  distcds 16897  ℝ^*_𝑠cxrs 17128  measurescmeas 32063  sitmcsitm 32195  sitgcsitg 32196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-1st 7804  df-2nd 7805  df-sitm 32198

This theorem is referenced by:  sitmfval  32217  sitmf  32219