Theorem measvun 29599

Description: The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)

Assertion

Ref	Expression
measvun	⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑀

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem measvun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1055	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2		measbase 29587	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
3		ismeas 29589	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)))))
4	2, 3	syl 17	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)))))
5	4	ibi 255	. . . 4 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥))))
6	5	simp3d 1068	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)))
7	6	3ad2ant1 1075	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)))
8		simp3 1056	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥))
9		breq1 4586	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10		disjeq1 4560	. . . . 5 ⊢ (𝑦 = 𝐴 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝐴 𝑥))
11	9, 10	anbi12d 743	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)))
12		unieq 4380	. . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
13	12	fveq2d 6107	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝐴))
14		esumeq1 29423	. . . . 5 ⊢ (𝑦 = 𝐴 → Σ*𝑥 ∈ 𝑦(𝑀‘𝑥) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))
15	13, 14	eqeq12d 2625	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥) ↔ (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥)))
16	11, 15	imbi12d 333	. . 3 ⊢ (𝑦 = 𝐴 → (((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)) ↔ ((𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥) → (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))))
17	16	rspcv 3278	. 2 ⊢ (𝐴 ∈ 𝒫 𝑆 → (∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)) → ((𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥) → (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))))
18	1, 7, 8, 17	syl3c 64	1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  Disj wdisj 4553   class class class wbr 4583  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957   ≼ cdom 7839  0cc0 9815  +∞cpnf 9950  [,]cicc 12049  Σ^*cesum 29416  sigAlgebracsiga 29497  measurescmeas 29585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-esum 29417  df-meas 29586

This theorem is referenced by:  measxun2  29600  measvunilem  29602  measssd  29605  measres  29612  measdivcstOLD  29614  measdivcst  29615  probcun  29807  totprobd  29815