Theorem sitmfval 30540

Description: Value of the integral distance between two simple functions. (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)
sitmfval.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
sitmfval.2	⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))

Assertion

Ref	Expression
sitmfval	⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)))

Proof of Theorem sitmfval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitmval.d	. . 3 ⊢ 𝐷 = (dist‘𝑊)
2		sitmval.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
3		sitmval.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
4	1, 2, 3	sitmval 30539	. 2 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘𝑓 𝐷𝑔))))
5		simprl 809	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹)
6		simprr 811	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺)
7	5, 6	oveq12d 6708	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 ∘𝑓 𝐷𝑔) = (𝐹 ∘𝑓 𝐷𝐺))
8	7	fveq2d 6233	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘𝑓 𝐷𝑔)) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)))
9		sitmfval.1	. 2 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
10		sitmfval.2	. 2 ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))
11		fvexd 6241	. 2 ⊢ (𝜑 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)) ∈ V)
12	4, 8, 9, 10, 11	ovmpt2d 6830	1 ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∪ cuni 4468  dom cdm 5143  ran crn 5144  ‘cfv 5926  (class class class)co 6690   ∘_𝑓 cof 6937  0cc0 9974  +∞cpnf 10109  [,]cicc 12216   ↾_s cress 15905  distcds 15997  ℝ^*_𝑠cxrs 16207  measurescmeas 30386  sitmcsitm 30518  sitgcsitg 30519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-1st 7210  df-2nd 7211  df-sitm 30521

This theorem is referenced by:  sitmcl  30541  sitmf  30542