Theorem adddmmbl 46754

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 4245	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
5		eqid 2740	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷))
6		ovexd 7483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (𝐶 + 𝐷) ∈ V)
7	1, 4, 5, 6	dmmptdff 45130	. 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 46273	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑆)
12	7, 11	eqeltrd 2844	1 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893  Vcvv 3488   ∩ cin 3975   ↦ cmpt 5249  dom cdm 5700  (class class class)co 7448   + caddc 11187  SAlgcsalg 46229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-salg 46230

This theorem is referenced by: (None)