Theorem meadjuni 46503

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 5092	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
4		disjeq1 5063	. . . . 5 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥))
5	3, 4	anbi12d 632	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥)))
6		unieq 4867	. . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋)
7	6	fveq2d 6826	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋))
8		reseq2 5922	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋))
9	8	fveq2d 6826	. . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋)))
10	7, 9	eqeq12d 2747	. . . 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 46497	. . . . 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 218	. . . 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 495	. . 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 46498	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
18		meadjuni.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
19	17, 18	ssexd 5260	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
20	18, 16	sseqtrdi 3970	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀)
21	19, 20	elpwd 4553	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀)
22	11, 15, 21	rspcdva 3573	. 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
23	1, 2, 22	mp2and 699	1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  ∪ cuni 4856  Disj wdisj 5056   class class class wbr 5089  dom cdm 5614   ↾ cres 5616  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ωcom 7796   ≼ cdom 8867  0cc0 11006  +∞cpnf 11143  [,]cicc 13248  SAlgcsalg 46354  Σ^{^}csumge0 46408  Meascmea 46495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-disj 5057  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-mea 46496

This theorem is referenced by:  meadjun  46508  meadjiun  46512