Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > saliincl | Structured version Visualization version GIF version |
Description: SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
saliincl.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
saliincl.kct | ⊢ (𝜑 → 𝐾 ≼ ω) |
saliincl.kn0 | ⊢ (𝜑 → 𝐾 ≠ ∅) |
saliincl.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) |
Ref | Expression |
---|---|
saliincl | ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | saliincl.e | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) | |
2 | elssuni 4860 | . . . . . . . 8 ⊢ (𝐸 ∈ 𝑆 → 𝐸 ⊆ ∪ 𝑆) | |
3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ⊆ ∪ 𝑆) |
4 | df-ss 3951 | . . . . . . 7 ⊢ (𝐸 ⊆ ∪ 𝑆 ↔ (𝐸 ∩ ∪ 𝑆) = 𝐸) | |
5 | 3, 4 | sylib 220 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝐸 ∩ ∪ 𝑆) = 𝐸) |
6 | 5 | eqcomd 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 = (𝐸 ∩ ∪ 𝑆)) |
7 | incom 4177 | . . . . . 6 ⊢ (𝐸 ∩ ∪ 𝑆) = (∪ 𝑆 ∩ 𝐸) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝐸 ∩ ∪ 𝑆) = (∪ 𝑆 ∩ 𝐸)) |
9 | dfin4 4243 | . . . . . 6 ⊢ (∪ 𝑆 ∩ 𝐸) = (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸)) | |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (∪ 𝑆 ∩ 𝐸) = (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸))) |
11 | 6, 8, 10 | 3eqtrd 2860 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 = (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸))) |
12 | 11 | iineq2dv 4936 | . . 3 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 = ∩ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸))) |
13 | saliincl.kn0 | . . . 4 ⊢ (𝜑 → 𝐾 ≠ ∅) | |
14 | iindif2 4991 | . . . 4 ⊢ (𝐾 ≠ ∅ → ∩ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸)) = (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸))) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ (∪ 𝑆 ∖ 𝐸)) = (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸))) |
16 | 12, 15 | eqtrd 2856 | . 2 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 = (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸))) |
17 | saliincl.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
18 | saliincl.kct | . . . 4 ⊢ (𝜑 → 𝐾 ≼ ω) | |
19 | 17 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝑆 ∈ SAlg) |
20 | saldifcl 42598 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | |
21 | 19, 1, 20 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
22 | 17, 18, 21 | saliuncl 42601 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
23 | saldifcl 42598 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) → (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸)) ∈ 𝑆) | |
24 | 17, 22, 23 | syl2anc 586 | . 2 ⊢ (𝜑 → (∪ 𝑆 ∖ ∪ 𝑘 ∈ 𝐾 (∪ 𝑆 ∖ 𝐸)) ∈ 𝑆) |
25 | 16, 24 | eqeltrd 2913 | 1 ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4831 ∪ ciun 4911 ∩ ciin 4912 class class class wbr 5058 ωcom 7574 ≼ cdom 8501 SAlgcsalg 42587 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-card 9362 df-acn 9365 df-salg 42588 |
This theorem is referenced by: iocborel 42633 hoimbllem 42906 iccvonmbllem 42954 salpreimagtge 42996 salpreimaltle 42997 smflimlem1 43041 smfsuplem1 43079 |
Copyright terms: Public domain | W3C validator |