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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > saliinclf | Structured version Visualization version GIF version |
Description: SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
Ref | Expression |
---|---|
saliinclf.1 | ⊢ Ⅎ𝑘𝜑 |
saliinclf.2 | ⊢ Ⅎ𝑘𝑆 |
saliinclf.3 | ⊢ Ⅎ𝑘𝐾 |
saliinclf.4 | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
saliinclf.5 | ⊢ (𝜑 → 𝐾 ≼ ω) |
saliinclf.6 | ⊢ (𝜑 → 𝐾 ≠ ∅) |
saliinclf.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) |
Ref | Expression |
---|---|
saliinclf | ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | saliinclf.1 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
2 | incom 4181 | . . . . 5 ⊢ (𝐸 ∩ ∪ 𝑆) = (∪ 𝑆 ∩ 𝐸) | |
3 | saliinclf.7 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) | |
4 | elssuni 4918 | . . . . . . 7 ⊢ (𝐸 ∈ 𝑆 → 𝐸 ⊆ ∪ 𝑆) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ⊆ ∪ 𝑆) |
6 | df-ss 3945 | . . . . . 6 ⊢ (𝐸 ⊆ ∪ 𝑆 ↔ (𝐸 ∩ ∪ 𝑆) = 𝐸) | |
7 | 5, 6 | sylib 217 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝐸 ∩ ∪ 𝑆) = 𝐸) |
8 | dfin4 4247 | . . . . . 6 ⊢ (∪ 𝑆 ∩ 𝐸) = (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸)) | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (∪ 𝑆 ∩ 𝐸) = (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸))) |
10 | 2, 7, 9 | 3eqtr3a 2795 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 = (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸))) |
11 | 1, 10 | iineq2d 4997 | . . 3 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 = ∩ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸))) |
12 | saliinclf.6 | . . . 4 ⊢ (𝜑 → 𝐾 ≠ ∅) | |
13 | saliinclf.3 | . . . . 5 ⊢ Ⅎ𝑘𝐾 | |
14 | saliinclf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝑆 | |
15 | 14 | nfuni 4892 | . . . . 5 ⊢ Ⅎ𝑘∪ 𝑆 |
16 | 13, 15 | iindif2f 43530 | . . . 4 ⊢ (𝐾 ≠ ∅ → ∩ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸)) = (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸))) |
17 | 12, 16 | syl 17 | . . 3 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸)) = (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸))) |
18 | 11, 17 | eqtrd 2771 | . 2 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 = (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸))) |
19 | saliinclf.4 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
20 | saliinclf.5 | . . . 4 ⊢ (𝜑 → 𝐾 ≼ ω) | |
21 | saldifcl 44713 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | |
22 | 19, 3, 21 | syl2an2r 683 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
23 | 1, 14, 13, 19, 20, 22 | saliunclf 44716 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
24 | saldifcl 44713 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) → (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸)) ∈ 𝑆) | |
25 | 19, 23, 24 | syl2anc 584 | . 2 ⊢ (𝜑 → (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸)) ∈ 𝑆) |
26 | 18, 25 | eqeltrd 2832 | 1 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2882 ≠ wne 2939 ∖ cdif 3925 ∩ cin 3927 ⊆ wss 3928 ∅c0 4302 ∪ cuni 4885 ∪ ciun 4974 ∩ ciin 4975 class class class wbr 5125 ωcom 7822 ≼ cdom 8903 SAlgcsalg 44702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-iin 4977 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-er 8670 df-map 8789 df-en 8906 df-dom 8907 df-card 9899 df-acn 9902 df-salg 44703 |
This theorem is referenced by: saliincl 44721 smfsupdmmbllem 45238 smfinfdmmbllem 45242 |
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