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Mathbox for Glauco Siliprandi |
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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 5965 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
3 | smfdivdmmbl2.9 | . . . . 5 ⊢ 𝐷 = {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} | |
4 | nfrab1 3454 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} | |
5 | 3, 4 | nfcxfr 2901 | . . . 4 ⊢ Ⅎ𝑥𝐷 |
6 | 2, 5 | nfin 4232 | . . 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 45212 | . 2 ⊢ dom 𝐻 = (dom 𝐹 ∩ 𝐷) |
10 | smfdivdmmbl2.4 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
11 | smfdivdmmbl2.5 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) | |
12 | 11 | fdmd 6747 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
13 | smfdivdmmbl2.7 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
14 | 12, 13 | eqeltrd 2839 | . . 3 ⊢ (𝜑 → dom 𝐹 ∈ 𝑆) |
15 | smfdivdmmbl2.8 | . . . . 5 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) | |
16 | 10, 15 | salrestss 46317 | . . . 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 2735 | . . . . . 6 ⊢ dom 𝐺 = dom 𝐺 | |
21 | 17, 18, 10, 19, 20 | smfpimne2 46796 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} ∈ (𝑆 ↾t dom 𝐺)) |
22 | 3, 21 | eqeltrid 2843 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝑆 ↾t dom 𝐺)) |
23 | 16, 22 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑆) |
24 | 10, 14, 23 | salincld 46308 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐷) ∈ 𝑆) |
25 | 9, 24 | eqeltrid 2843 | 1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ≠ wne 2938 {crab 3433 ∩ cin 3962 ↦ cmpt 5231 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 0cc0 11153 / cdiv 11918 ↾t crest 17467 SAlgcsalg 46264 SMblFncsmblfn 46651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-ioo 13388 df-ico 13390 df-fl 13829 df-rest 17469 df-salg 46265 df-smblfn 46652 |
This theorem is referenced by: (None) |
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