Theorem sitmfval 34311

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 34310	. 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 7431	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 ∘f 𝐷𝑔) = (𝐹 ∘f 𝐷𝐺))
8	7	fveq2d 6890	. 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 6901	. 2 ⊢ (𝜑 → (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺)) ∈ V)
12	4, 8, 9, 10, 11	ovmpod 7567	1 ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ∪ cuni 4887  dom cdm 5665  ran crn 5666  ‘cfv 6541  (class class class)co 7413   ∘_f cof 7677  0cc0 11137  +∞cpnf 11274  [,]cicc 13372   ↾_s cress 17252  distcds 17282  ℝ^*_𝑠cxrs 17516  measurescmeas 34155  sitmcsitm 34289  sitgcsitg 34290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-1st 7996  df-2nd 7997  df-sitm 34292

This theorem is referenced by:  sitmcl  34312  sitmf  34313