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| Mirrors > Home > MPE Home > Th. List > Mathboxes > smfdivdmmbl2 | Structured version Visualization version GIF version | ||
| Description: If a functions and a sigma-measurable function have domains in the sigma-algebra, the domain of the division of the two functions is in the sigma-algebra. This is the third statement of Proposition 121H of [Fremlin1] p. 39 . Note: While the theorem in the book assumes both functions are sigma-measurable, this assumption is unnecessary for the part concerning their division, for the function at the numerator. It is required only for the function at the denominator. (Contributed by Glauco Siliprandi, 5-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| smfdivdmmbl2.1 | ⊢ Ⅎ𝑥𝜑 | 
| smfdivdmmbl2.2 | ⊢ Ⅎ𝑥𝐹 | 
| smfdivdmmbl2.3 | ⊢ Ⅎ𝑥𝐺 | 
| smfdivdmmbl2.4 | ⊢ (𝜑 → 𝑆 ∈ SAlg) | 
| smfdivdmmbl2.5 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) | 
| smfdivdmmbl2.6 | ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | 
| smfdivdmmbl2.7 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) | 
| smfdivdmmbl2.8 | ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) | 
| smfdivdmmbl2.9 | ⊢ 𝐷 = {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} | 
| smfdivdmmbl2.10 | ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ 𝐷) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))) | 
| Ref | Expression | 
|---|---|
| smfdivdmmbl2 | ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | smfdivdmmbl2.2 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
| 2 | 1 | nfdm 5962 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 | 
| 3 | smfdivdmmbl2.9 | . . . . 5 ⊢ 𝐷 = {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} | |
| 4 | nfrab1 3457 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} | |
| 5 | 3, 4 | nfcxfr 2903 | . . . 4 ⊢ Ⅎ𝑥𝐷 | 
| 6 | 2, 5 | nfin 4224 | . . 3 ⊢ Ⅎ𝑥(dom 𝐹 ∩ 𝐷) | 
| 7 | ovex 7464 | . . 3 ⊢ ((𝐹‘𝑥) / (𝐺‘𝑥)) ∈ V | |
| 8 | smfdivdmmbl2.10 | . . 3 ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ 𝐷) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))) | |
| 9 | 6, 7, 8 | dmmptif 45273 | . 2 ⊢ dom 𝐻 = (dom 𝐹 ∩ 𝐷) | 
| 10 | smfdivdmmbl2.4 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 11 | smfdivdmmbl2.5 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) | |
| 12 | 11 | fdmd 6746 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴) | 
| 13 | smfdivdmmbl2.7 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 14 | 12, 13 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → dom 𝐹 ∈ 𝑆) | 
| 15 | smfdivdmmbl2.8 | . . . . 5 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) | |
| 16 | 10, 15 | salrestss 46376 | . . . 4 ⊢ (𝜑 → (𝑆 ↾t dom 𝐺) ⊆ 𝑆) | 
| 17 | smfdivdmmbl2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
| 18 | smfdivdmmbl2.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐺 | |
| 19 | smfdivdmmbl2.6 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | |
| 20 | eqid 2737 | . . . . . 6 ⊢ dom 𝐺 = dom 𝐺 | |
| 21 | 17, 18, 10, 19, 20 | smfpimne2 46855 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} ∈ (𝑆 ↾t dom 𝐺)) | 
| 22 | 3, 21 | eqeltrid 2845 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝑆 ↾t dom 𝐺)) | 
| 23 | 16, 22 | sseldd 3984 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | 
| 24 | 10, 14, 23 | salincld 46367 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐷) ∈ 𝑆) | 
| 25 | 9, 24 | eqeltrid 2845 | 1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 {crab 3436 ∩ cin 3950 ↦ cmpt 5225 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 / cdiv 11920 ↾t crest 17465 SAlgcsalg 46323 SMblFncsmblfn 46710 | 
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 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-iin 4994 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-se 5638 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-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 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-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-ioo 13391 df-ico 13393 df-fl 13832 df-rest 17467 df-salg 46324 df-smblfn 46711 | 
| This theorem is referenced by: (None) | 
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