Theorem sitmval 34763

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 3478	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑊 ∈ V)
4		sitmval.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
5		oveq1 7423	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
6	5	dmeqd 5897	. . . 4 ⊢ (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7		fveq2 6885	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8	7	ofeqd 7682	. . . . . 6 ⊢ (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
9	8	oveqd 7433	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (dist‘𝑤)𝑔) = (𝑓 ∘f (dist‘𝑊)𝑔))
10	9	fveq2d 6889	. . . 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 7491	. . 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 7424	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
13	12	dmeqd 5897	. . . 4 ⊢ (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
14		oveq2 7424	. . . . 5 ⊢ (𝑚 = 𝑀 → ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀))
15		sitmval.d	. . . . . . . 8 ⊢ 𝐷 = (dist‘𝑊)
16	15	eqcomi 2774	. . . . . . 7 ⊢ (dist‘𝑊) = 𝐷
17		ofeq 7683	. . . . . . 7 ⊢ ((dist‘𝑊) = 𝐷 → ∘f (dist‘𝑊) = ∘f 𝐷)
18	16, 17	mp1i 14	. . . . . 6 ⊢ (𝑚 = 𝑀 → ∘f (dist‘𝑊) = ∘f 𝐷)
19	18	oveqd 7433	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑓 ∘f (dist‘𝑊)𝑔) = (𝑓 ∘f 𝐷𝑔))
20	14, 19	fveq12d 6892	. . . 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 7491	. . 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 34745	. . 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 7449	. . . . 5 ⊢ (𝑊sitg𝑀) ∈ V
24	23	dmex 7908	. . . 4 ⊢ dom (𝑊sitg𝑀) ∈ V
25	24, 24	mpoex 8078	. . 3 ⊢ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))) ∈ V
26	11, 21, 22, 25	ovmpo 7576	. 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 596	1 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∪ cuni 4874  dom cdm 5663  ran crn 5664  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418   ∘_f cof 7678  0cc0 11111  +∞cpnf 11251  [,]cicc 13386   ↾_s cress 17307  distcds 17336  ℝ^*_𝑠cxrs 17571  measurescmeas 34609  sitmcsitm 34742  sitgcsitg 34743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7680  df-1st 7988  df-2nd 7989  df-sitm 34745

This theorem is used by:  sitmfval  34764  sitmf  34766