Theorem meadjuni 46078

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	. 2 ⊢ (𝜑 → 𝑋 ≼ ω)
2		meadjuni.dj	. 2 ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥)
3		breq1 5156	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
4		disjeq1 5125	. . . . 5 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥))
5	3, 4	anbi12d 630	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥)))
6		unieq 4924	. . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋)
7	6	fveq2d 6905	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋))
8		reseq2 5984	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋))
9	8	fveq2d 6905	. . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋)))
10	7, 9	eqeq12d 2742	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
11	5, 10	imbi12d 343	. . 3 ⊢ (𝑦 = 𝑋 → (((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))) ↔ ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))))
12		meadjuni.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Meas)
13		ismea 46072	. . . . 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	12, 13	sylib 217	. . . 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 494	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))))
16		meadjuni.s	. . . . . 6 ⊢ 𝑆 = dom 𝑀
17	12, 16	dmmeasal 46073	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
18		meadjuni.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
19	17, 18	ssexd 5329	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
20	18, 16	sseqtrdi 4030	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀)
21	19, 20	elpwd 4613	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀)
22	11, 15, 21	rspcdva 3609	. 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
23	1, 2, 22	mp2and 697	1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051  Vcvv 3462   ⊆ wss 3947  ∅c0 4325  𝒫 cpw 4607  ∪ cuni 4913  Disj wdisj 5118   class class class wbr 5153  dom cdm 5682   ↾ cres 5684  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424  ωcom 7876   ≼ cdom 8972  0cc0 11158  +∞cpnf 11295  [,]cicc 13381  SAlgcsalg 45929  Σ^{^}csumge0 45983  Meascmea 46070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-disj 5119  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-mea 46071

This theorem is referenced by:  meadjun  46083  meadjiun  46087