Theorem measvun 34173

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 34161	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
3		ismeas 34163	. . . . . 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 5169	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10		disjeq1 5140	. . . . 5 ⊢ (𝑦 = 𝐴 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝐴 𝑥))
11	9, 10	anbi12d 631	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)))
12		unieq 4942	. . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
13	12	fveq2d 6924	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝐴))
14		esumeq1 33998	. . . . 5 ⊢ (𝑦 = 𝐴 → Σ*𝑥 ∈ 𝑦(𝑀‘𝑥) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))
15	13, 14	eqeq12d 2756	. . . 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 3631	. 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931  Disj wdisj 5133   class class class wbr 5166  ran crn 5701  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ωcom 7903   ≼ cdom 9001  0cc0 11184  +∞cpnf 11321  [,]cicc 13410  Σ^*cesum 33991  sigAlgebracsiga 34072  measurescmeas 34159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-esum 33992  df-meas 34160

This theorem is referenced by:  measxun2  34174  measvunilem  34176  measssd  34179  measres  34186  measdivcst  34188  measdivcstALTV  34189  probcun  34383  totprobd  34391