Theorem adddmmbl2 46816

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 5897	. . . . 5 ⊢ Ⅎ𝑥dom 𝐹
3		adddmmbl2.2	. . . . . 6 ⊢ Ⅎ𝑥𝐺
4	3	nfdm 5897	. . . . 5 ⊢ Ⅎ𝑥dom 𝐺
5	2, 4	nfin 4177	. . . 4 ⊢ Ⅎ𝑥(dom 𝐹 ∩ dom 𝐺)
6		ovex 7386	. . . 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 45244	. . 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 46334	. 2 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ 𝑆)
14	9, 13	eqeltrd 2828	1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2876   ∩ cin 3904   ↦ cmpt 5176  dom cdm 5623  ‘cfv 6486  (class class class)co 7353   + caddc 11031  SAlgcsalg 46290

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-salg 46291

This theorem is referenced by: (None)