Theorem adddmmbl2 46849

Description: If two functions have domains in the sigma-algebra, the domain of their addition also belongs to the sigma-algebra. This is the first statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their addition. (Contributed by Glauco Siliprandi, 30-Dec-2024.)

Hypotheses

Ref	Expression
adddmmbl2.1	⊢ Ⅎ𝑥𝐹
adddmmbl2.2	⊢ Ⅎ𝑥𝐺
adddmmbl2.3	⊢ (𝜑 → 𝑆 ∈ SAlg)
adddmmbl2.4	⊢ (𝜑 → dom 𝐹 ∈ 𝑆)
adddmmbl2.5	⊢ (𝜑 → dom 𝐺 ∈ 𝑆)
adddmmbl2.6	⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) + (𝐺‘𝑥)))

Assertion

Ref	Expression
adddmmbl2	⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

Proof of Theorem adddmmbl2

Step	Hyp	Ref	Expression
1		adddmmbl2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
2	1	nfdm 5962	. . . . 5 ⊢ Ⅎ𝑥dom 𝐹
3		adddmmbl2.2	. . . . . 6 ⊢ Ⅎ𝑥𝐺
4	3	nfdm 5962	. . . . 5 ⊢ Ⅎ𝑥dom 𝐺
5	2, 4	nfin 4224	. . . 4 ⊢ Ⅎ𝑥(dom 𝐹 ∩ dom 𝐺)
6		ovex 7464	. . . 4 ⊢ ((𝐹‘𝑥) + (𝐺‘𝑥)) ∈ V
7		adddmmbl2.6	. . . 4 ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) + (𝐺‘𝑥)))
8	5, 6, 7	dmmptif 45273	. . 3 ⊢ dom 𝐻 = (dom 𝐹 ∩ dom 𝐺)
9	8	a1i 11	. 2 ⊢ (𝜑 → dom 𝐻 = (dom 𝐹 ∩ dom 𝐺))
10		adddmmbl2.3	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11		adddmmbl2.4	. . 3 ⊢ (𝜑 → dom 𝐹 ∈ 𝑆)
12		adddmmbl2.5	. . 3 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)
13	10, 11, 12	salincld 46367	. 2 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ 𝑆)
14	9, 13	eqeltrd 2841	1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890   ∩ cin 3950   ↦ cmpt 5225  dom cdm 5685  ‘cfv 6561  (class class class)co 7431   + caddc 11158  SAlgcsalg 46323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-salg 46324

This theorem is referenced by: (None)