Theorem meadjuni 45159

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 5150	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
4		disjeq1 5119	. . . . 5 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥))
5	3, 4	anbi12d 631	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥)))
6		unieq 4918	. . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋)
7	6	fveq2d 6892	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋))
8		reseq2 5974	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋))
9	8	fveq2d 6892	. . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋)))
10	7, 9	eqeq12d 2748	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
11	5, 10	imbi12d 344	. . 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 45153	. . . . 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 496	. . 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 45154	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
18		meadjuni.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
19	17, 18	ssexd 5323	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
20	18, 16	sseqtrdi 4031	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀)
21	19, 20	elpwd 4607	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀)
22	11, 15, 21	rspcdva 3613	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  ∪ cuni 4907  Disj wdisj 5112   class class class wbr 5147  dom cdm 5675   ↾ cres 5677  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  ωcom 7851   ≼ cdom 8933  0cc0 11106  +∞cpnf 11241  [,]cicc 13323  SAlgcsalg 45010  Σ^{^}csumge0 45064  Meascmea 45151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-mea 45152

This theorem is referenced by:  meadjun  45164  meadjiun  45168