Theorem sitmval 35085

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 3475	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑊 ∈ V)
4		sitmval.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
5		oveq1 7456	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤sitg𝑚) = (𝑊sitg𝑚))
6	5	dmeqd 5897	. . . 4 ⊢ (𝑤 = 𝑊 → dom (𝑤sitg𝑚) = dom (𝑊sitg𝑚))
7		fveq2 6914	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
8	7	ofeqd 7725	. . . . . 6 ⊢ (𝑤 = 𝑊 → ∘f (dist‘𝑤) = ∘f (dist‘𝑊))
9	8	oveqd 7466	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (dist‘𝑤)𝑔) = (𝑓 ∘f (dist‘𝑊)𝑔))
10	9	fveq2d 6918	. . . 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 7524	. . 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 7457	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑊sitg𝑚) = (𝑊sitg𝑀))
13	12	dmeqd 5897	. . . 4 ⊢ (𝑚 = 𝑀 → dom (𝑊sitg𝑚) = dom (𝑊sitg𝑀))
14		oveq2 7457	. . . . 5 ⊢ (𝑚 = 𝑀 → ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚) = ((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀))
15		sitmval.d	. . . . . . . 8 ⊢ 𝐷 = (dist‘𝑊)
16	15	eqcomi 2772	. . . . . . 7 ⊢ (dist‘𝑊) = 𝐷
17		ofeq 7726	. . . . . . 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 7466	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑓 ∘f (dist‘𝑊)𝑔) = (𝑓 ∘f 𝐷𝑔))
20	14, 19	fveq12d 6921	. . . 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 7524	. . 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 35067	. . 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 7482	. . . . 5 ⊢ (𝑊sitg𝑀) ∈ V
24	23	dmex 7953	. . . 4 ⊢ dom (𝑊sitg𝑀) ∈ V
25	24, 24	mpoex 8125	. . 3 ⊢ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))) ∈ V
26	11, 21, 22, 25	ovmpo 7610	. 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 2147  Vcvv 3454  ∪ cuni 4870  dom cdm 5656  ran crn 5657  ‘cfv 6564  (class class class)co 7449   ∈ cmpo 7451   ∘_f cof 7721  0cc0 11237  +∞cpnf 11377  [,]cicc 13517   ↾_s cress 17449  distcds 17478  ℝ^*_𝑠cxrs 17713  measurescmeas 34932  sitmcsitm 35064  sitgcsitg 35065

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-ov 7452  df-oprab 7453  df-mpo 7454  df-of 7723  df-1st 8033  df-2nd 8034  df-sitm 35067

This theorem is used by:  sitmfval  35086  sitmf  35088