Theorem measvun 34240

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 1137	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2		measbase 34228	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
3		ismeas 34230	. . . . . 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 267	. . . 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 1144	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)))
7	6	3ad2ant1 1133	. 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 1138	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥))
9		breq1 5122	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10		disjeq1 5093	. . . . 5 ⊢ (𝑦 = 𝐴 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝐴 𝑥))
11	9, 10	anbi12d 632	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)))
12		unieq 4894	. . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
13	12	fveq2d 6880	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝐴))
14		esumeq1 34065	. . . . 5 ⊢ (𝑦 = 𝐴 → Σ*𝑥 ∈ 𝑦(𝑀‘𝑥) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))
15	13, 14	eqeq12d 2751	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥) ↔ (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥)))
16	11, 15	imbi12d 344	. . 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 3597	. 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 66	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∅c0 4308  𝒫 cpw 4575  ∪ cuni 4883  Disj wdisj 5086   class class class wbr 5119  ran crn 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  ωcom 7861   ≼ cdom 8957  0cc0 11129  +∞cpnf 11266  [,]cicc 13365  Σ^*cesum 34058  sigAlgebracsiga 34139  measurescmeas 34226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-disj 5087  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-esum 34059  df-meas 34227

This theorem is referenced by:  measxun2  34241  measvunilem  34243  measssd  34246  measres  34253  measdivcst  34255  measdivcstALTV  34256  probcun  34450  totprobd  34458