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| Mirrors > Home > MPE Home > Th. List > Mathboxes > adddmmbl2 | Structured version Visualization version GIF version | ||
| 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.) | 
| 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))) | 
| Ref | Expression | 
|---|---|
| adddmmbl2 | ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | adddmmbl2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
| 2 | 1 | nfdm 5962 | . . . . 5 ⊢ Ⅎ𝑥dom 𝐹 | 
| 3 | adddmmbl2.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐺 | |
| 4 | 3 | nfdm 5962 | . . . . 5 ⊢ Ⅎ𝑥dom 𝐺 | 
| 5 | 2, 4 | nfin 4224 | . . . 4 ⊢ Ⅎ𝑥(dom 𝐹 ∩ dom 𝐺) | 
| 6 | ovex 7464 | . . . 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 45273 | . . 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 46367 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ 𝑆) | 
| 14 | 9, 13 | eqeltrd 2841 | 1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 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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