Theorem adddmmbl2 47192

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 5908	. . . . 5 ⊢ Ⅎ𝑥dom 𝐹
3		adddmmbl2.2	. . . . . 6 ⊢ Ⅎ𝑥𝐺
4	3	nfdm 5908	. . . . 5 ⊢ Ⅎ𝑥dom 𝐺
5	2, 4	nfin 4178	. . . 4 ⊢ Ⅎ𝑥(dom 𝐹 ∩ dom 𝐺)
6		ovex 7401	. . . 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 45624	. . 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 46710	. 2 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ 𝑆)
14	9, 13	eqeltrd 2837	1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884   ∩ cin 3902   ↦ cmpt 5181  dom cdm 5632  ‘cfv 6500  (class class class)co 7368   + caddc 11041  SAlgcsalg 46666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-salg 46667

This theorem is referenced by: (None)