Theorem meadjuni 39350

Description: The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
meadjuni.m	⊢ (𝜑 → 𝑀 ∈ Meas)
meadjuni.s	⊢ 𝑆 = dom 𝑀
meadjuni.x	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
meadjuni.cnb	⊢ (𝜑 → 𝑋 ≼ ω)
meadjuni.dj	⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥)

Assertion

Ref	Expression
meadjuni	⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))

Distinct variable group:   𝑥,𝑋

Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)   𝑀(𝑥)

Proof of Theorem meadjuni

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		meadjuni.cnb	. . 3 ⊢ (𝜑 → 𝑋 ≼ ω)
2		meadjuni.dj	. . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥)
3	1, 2	jca 553	. 2 ⊢ (𝜑 → (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥))
4		meadjuni.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
5		meadjuni.s	. . . . 5 ⊢ 𝑆 = dom 𝑀
6	4, 5	syl6sseq 3614	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀)
7		meadjuni.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Meas)
8	7, 5	dmmeasal 39345	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ SAlg)
9	8, 4	ssexd 4733	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
10		elpwg 4116	. . . . 5 ⊢ (𝑋 ∈ V → (𝑋 ∈ 𝒫 dom 𝑀 ↔ 𝑋 ⊆ dom 𝑀))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝒫 dom 𝑀 ↔ 𝑋 ⊆ dom 𝑀))
12	6, 11	mpbird 246	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀)
13		ismea 39344	. . . . 5 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)))))
14	7, 13	sylib 207	. . . 4 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)))))
15	14	simprd 478	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))))
16		breq1 4586	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
17		disjeq1 4560	. . . . . 6 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥))
18	16, 17	anbi12d 743	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥)))
19		unieq 4380	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋)
20	19	fveq2d 6107	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋))
21		reseq2 5312	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋))
22	21	fveq2d 6107	. . . . . 6 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋)))
23	20, 22	eqeq12d 2625	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
24	18, 23	imbi12d 333	. . . 4 ⊢ (𝑦 = 𝑋 → (((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))) ↔ ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))))
25	24	rspcva 3280	. . 3 ⊢ ((𝑋 ∈ 𝒫 dom 𝑀 ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)))) → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
26	12, 15, 25	syl2anc 691	. 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
27	3, 26	mpd 15	1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  Disj wdisj 4553   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957   ≼ cdom 7839  0cc0 9815  +∞cpnf 9950  [,]cicc 12049  SAlgcsalg 39204  Σ^{^}csumge0 39255  Meascmea 39342

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-reu 2903  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-iun 4457  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-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-mea 39343

This theorem is referenced by:  meadjun  39355  meadjiun  39359