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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 𝑥𝐹
adddmmbl2.2 𝑥𝐺
adddmmbl2.3 (𝜑𝑆 ∈ SAlg)
adddmmbl2.4 (𝜑 → dom 𝐹𝑆)
adddmmbl2.5 (𝜑 → dom 𝐺𝑆)
adddmmbl2.6 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) + (𝐺𝑥)))
Assertion
Ref Expression
adddmmbl2 (𝜑 → dom 𝐻𝑆)

Proof of Theorem adddmmbl2
StepHypRef Expression
1 adddmmbl2.1 . . . . . 6 𝑥𝐹
21nfdm 5962 . . . . 5 𝑥dom 𝐹
3 adddmmbl2.2 . . . . . 6 𝑥𝐺
43nfdm 5962 . . . . 5 𝑥dom 𝐺
52, 4nfin 4224 . . . 4 𝑥(dom 𝐹 ∩ dom 𝐺)
6 ovex 7464 . . . 4 ((𝐹𝑥) + (𝐺𝑥)) ∈ V
7 adddmmbl2.6 . . . 4 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) + (𝐺𝑥)))
85, 6, 7dmmptif 45273 . . 3 dom 𝐻 = (dom 𝐹 ∩ dom 𝐺)
98a1i 11 . 2 (𝜑 → dom 𝐻 = (dom 𝐹 ∩ dom 𝐺))
10 adddmmbl2.3 . . 3 (𝜑𝑆 ∈ SAlg)
11 adddmmbl2.4 . . 3 (𝜑 → dom 𝐹𝑆)
12 adddmmbl2.5 . . 3 (𝜑 → dom 𝐺𝑆)
1310, 11, 12salincld 46367 . 2 (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ 𝑆)
149, 13eqeltrd 2841 1 (𝜑 → dom 𝐻𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wnfc 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)
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