![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sigaclcu2 | Structured version Visualization version GIF version |
Description: A sigma-algebra is closed under countable union - indexing on ℕ (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
sigaclcu2 | ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ ℕ 𝐴 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiun2g 5026 | . . 3 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 → ∪ 𝑘 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}) |
3 | simpl 482 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
4 | abid 2707 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ↔ ∃𝑘 ∈ ℕ 𝑥 = 𝐴) | |
5 | eleq1a 2822 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → (𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) | |
6 | 5 | ralimi 3077 | . . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 → ∀𝑘 ∈ ℕ (𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) |
7 | r19.23v 3176 | . . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ (𝑥 = 𝐴 → 𝑥 ∈ 𝑆) ↔ (∃𝑘 ∈ ℕ 𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) | |
8 | 6, 7 | sylib 217 | . . . . . . . . 9 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 → (∃𝑘 ∈ ℕ 𝑥 = 𝐴 → 𝑥 ∈ 𝑆)) |
9 | 8 | imp 406 | . . . . . . . 8 ⊢ ((∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆 ∧ ∃𝑘 ∈ ℕ 𝑥 = 𝐴) → 𝑥 ∈ 𝑆) |
10 | 9 | adantll 711 | . . . . . . 7 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) ∧ ∃𝑘 ∈ ℕ 𝑥 = 𝐴) → 𝑥 ∈ 𝑆) |
11 | 4, 10 | sylan2b 593 | . . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) ∧ 𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}) → 𝑥 ∈ 𝑆) |
12 | 11 | ralrimiva 3140 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∀𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}𝑥 ∈ 𝑆) |
13 | nfab1 2899 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} | |
14 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑥𝑆 | |
15 | 13, 14 | dfss3f 3968 | . . . . 5 ⊢ ({𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆 ↔ ∀𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴}𝑥 ∈ 𝑆) |
16 | 12, 15 | sylibr 233 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆) |
17 | elpw2g 5337 | . . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ({𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆 ↔ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆)) | |
18 | 17 | adantr 480 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ({𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆 ↔ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ⊆ 𝑆)) |
19 | 16, 18 | mpbird 257 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆) |
20 | nnct 13952 | . . . 4 ⊢ ℕ ≼ ω | |
21 | abrexct 32448 | . . . 4 ⊢ (ℕ ≼ ω → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ≼ ω) | |
22 | 20, 21 | mp1i 13 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ≼ ω) |
23 | sigaclcu 33645 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝒫 𝑆 ∧ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ≼ ω) → ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝑆) | |
24 | 3, 19, 22, 23 | syl3anc 1368 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ {𝑥 ∣ ∃𝑘 ∈ ℕ 𝑥 = 𝐴} ∈ 𝑆) |
25 | 2, 24 | eqeltrd 2827 | 1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ ℕ 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 ∀wral 3055 ∃wrex 3064 ⊆ wss 3943 𝒫 cpw 4597 ∪ cuni 4902 ∪ ciun 4990 class class class wbr 5141 ran crn 5670 ωcom 7852 ≼ cdom 8939 ℕcn 12216 sigAlgebracsiga 33636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-siga 33637 |
This theorem is referenced by: sigaclfu2 33649 sigaclcu3 33650 measiun 33746 |
Copyright terms: Public domain | W3C validator |