Theorem adddmmbl 47547

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
adddmmbl.1	⊢ Ⅎ𝑥𝜑
adddmmbl.2	⊢ Ⅎ𝑥𝐴
adddmmbl.3	⊢ Ⅎ𝑥𝐵
adddmmbl.4	⊢ (𝜑 → 𝑆 ∈ SAlg)
adddmmbl.5	⊢ (𝜑 → 𝐴 ∈ 𝑆)
adddmmbl.6	⊢ (𝜑 → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
adddmmbl	⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) ∈ 𝑆)

Proof of Theorem adddmmbl

Step	Hyp	Ref	Expression
1		adddmmbl.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		adddmmbl.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		adddmmbl.3	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfin 4177	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
5		eqid 2763	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷))
6		ovexd 7445	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (𝐶 + 𝐷) ∈ V)
7	1, 4, 5, 6	dmmptdff 45939	. 2 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) = (𝐴 ∩ 𝐵))
8		adddmmbl.4	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
9		adddmmbl.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
10		adddmmbl.6	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
11	8, 9, 10	salincld 47066	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑆)
12	7, 11	eqeltrd 2863	1 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1813   ∈ wcel 2143  Ⅎwnfc 2910  Vcvv 3455   ∩ cin 3904   ↦ cmpt 5192  dom cdm 5661  (class class class)co 7410   + caddc 11098  SAlgcsalg 47022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  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 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-salg 47023

This theorem is referenced by: (None)