Theorem sitmfval 34374

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 34373	. 2 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))
5		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹)
6		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺)
7	5, 6	oveq12d 7373	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 ∘f 𝐷𝑔) = (𝐹 ∘f 𝐷𝐺))
8	7	fveq2d 6835	. 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 6846	. 2 ⊢ (𝜑 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺)) ∈ V)
12	4, 8, 9, 10, 11	ovmpod 7507	1 ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ∪ cuni 4860  dom cdm 5621  ran crn 5622  ‘cfv 6489  (class class class)co 7355   ∘_f cof 7617  0cc0 11016  +∞cpnf 11153  [,]cicc 13258   ↾_s cress 17151  distcds 17180  ℝ^*_𝑠cxrs 17414  measurescmeas 34219  sitmcsitm 34352  sitgcsitg 34353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-1st 7930  df-2nd 7931  df-sitm 34355

This theorem is referenced by:  sitmcl  34375  sitmf  34376