Theorem meadjuni 47062

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 5116	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
4		disjeq1 5087	. . . . 5 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥))
5	3, 4	anbi12d 643	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥)))
6		unieq 4887	. . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋)
7	6	fveq2d 6886	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋))
8		reseq2 5974	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋))
9	8	fveq2d 6886	. . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋)))
10	7, 9	eqeq12d 2785	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
11	5, 10	imbi12d 347	. . 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 47056	. . . . 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 221	. . . 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 500	. . 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 47057	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
18		meadjuni.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
19	17, 18	ssexd 5295	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
20	18, 16	sseqtrdi 3985	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀)
21	19, 20	elpwd 4573	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀)
22	11, 15, 21	rspcdva 3591	. 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))
23	1, 2, 22	mp2and 711	1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876  Disj wdisj 5080   class class class wbr 5113  dom cdm 5662   ↾ cres 5664  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ωcom 7861   ≼ cdom 8940  0cc0 11099  +∞cpnf 11239  [,]cicc 13374  SAlgcsalg 46913  Σ^{^}csumge0 46967  Meascmea 47054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-disj 5081  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-mea 47055

This theorem is referenced by:  meadjun  47067  meadjiun  47071