Theorem measvun 33144

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 1138	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2		measbase 33132	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
3		ismeas 33134	. . . . . 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 1145	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)))
7	6	3ad2ant1 1134	. 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 1139	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥))
9		breq1 5149	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10		disjeq1 5118	. . . . 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 4917	. . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
13	12	fveq2d 6891	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝐴))
14		esumeq1 32969	. . . . 5 ⊢ (𝑦 = 𝐴 → Σ*𝑥 ∈ 𝑦(𝑀‘𝑥) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥))
15	13, 14	eqeq12d 2749	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥) ↔ (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥)))
16	11, 15	imbi12d 345	. . 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 3607	. 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 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∅c0 4320  𝒫 cpw 4600  ∪ cuni 4906  Disj wdisj 5111   class class class wbr 5146  ran crn 5675  ⟶wf 6535  ‘cfv 6539  (class class class)co 7403  ωcom 7849   ≼ cdom 8932  0cc0 11105  +∞cpnf 11240  [,]cicc 13322  Σ^*cesum 32962  sigAlgebracsiga 33043  measurescmeas 33130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-disj 5112  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7406  df-esum 32963  df-meas 33131

This theorem is referenced by:  measxun2  33145  measvunilem  33147  measssd  33150  measres  33157  measdivcst  33159  measdivcstALTV  33160  probcun  33354  totprobd  33362