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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sxbrsigalem1 | Structured version Visualization version GIF version |
Description: The Borel algebra on (ℝ × ℝ) is a subset of the sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
Ref | Expression |
---|---|
sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
dya2ioc.2 | ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) |
Ref | Expression |
---|---|
sxbrsigalem1 | ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sxbrsiga.0 | . . . 4 ⊢ 𝐽 = (topGen‘ran (,)) | |
2 | dya2ioc.1 | . . . 4 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) | |
3 | dya2ioc.2 | . . . 4 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) | |
4 | 1, 2, 3 | dya2iocucvr 31178 | . . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ) |
5 | retop 23067 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | 1, 5 | eqeltri 2859 | . . . 4 ⊢ 𝐽 ∈ Top |
7 | uniretop 23068 | . . . . 5 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
8 | 1 | unieqi 4719 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
9 | 7, 8 | eqtr4i 2802 | . . . 4 ⊢ ℝ = ∪ 𝐽 |
10 | 6, 6, 9, 9 | txunii 21899 | . . 3 ⊢ (ℝ × ℝ) = ∪ (𝐽 ×t 𝐽) |
11 | 4, 10 | eqtr2i 2800 | . 2 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ ran 𝑅 |
12 | 1, 2, 3 | dya2iocuni 31177 | . . . 4 ⊢ (𝑥 ∈ (𝐽 ×t 𝐽) → ∃𝑦 ∈ 𝒫 ran 𝑅∪ 𝑦 = 𝑥) |
13 | simpr 477 | . . . . . 6 ⊢ ((𝑦 ∈ 𝒫 ran 𝑅 ∧ ∪ 𝑦 = 𝑥) → ∪ 𝑦 = 𝑥) | |
14 | 1, 2, 3 | dya2iocct 31174 | . . . . . . . . 9 ⊢ ran 𝑅 ≼ ω |
15 | ctex 8317 | . . . . . . . . 9 ⊢ (ran 𝑅 ≼ ω → ran 𝑅 ∈ V) | |
16 | 14, 15 | mp1i 13 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → ran 𝑅 ∈ V) |
17 | elpwi 4430 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → 𝑦 ⊆ ran 𝑅) | |
18 | ssct 8390 | . . . . . . . . 9 ⊢ ((𝑦 ⊆ ran 𝑅 ∧ ran 𝑅 ≼ ω) → 𝑦 ≼ ω) | |
19 | 17, 14, 18 | sylancl 577 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → 𝑦 ≼ ω) |
20 | elsigagen2 31043 | . . . . . . . 8 ⊢ ((ran 𝑅 ∈ V ∧ 𝑦 ⊆ ran 𝑅 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ (sigaGen‘ran 𝑅)) | |
21 | 16, 17, 19, 20 | syl3anc 1351 | . . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → ∪ 𝑦 ∈ (sigaGen‘ran 𝑅)) |
22 | 21 | adantr 473 | . . . . . 6 ⊢ ((𝑦 ∈ 𝒫 ran 𝑅 ∧ ∪ 𝑦 = 𝑥) → ∪ 𝑦 ∈ (sigaGen‘ran 𝑅)) |
23 | 13, 22 | eqeltrrd 2864 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 ran 𝑅 ∧ ∪ 𝑦 = 𝑥) → 𝑥 ∈ (sigaGen‘ran 𝑅)) |
24 | 23 | rexlimiva 3223 | . . . 4 ⊢ (∃𝑦 ∈ 𝒫 ran 𝑅∪ 𝑦 = 𝑥 → 𝑥 ∈ (sigaGen‘ran 𝑅)) |
25 | 12, 24 | syl 17 | . . 3 ⊢ (𝑥 ∈ (𝐽 ×t 𝐽) → 𝑥 ∈ (sigaGen‘ran 𝑅)) |
26 | 25 | ssriv 3861 | . 2 ⊢ (𝐽 ×t 𝐽) ⊆ (sigaGen‘ran 𝑅) |
27 | 14, 15 | ax-mp 5 | . 2 ⊢ ran 𝑅 ∈ V |
28 | sigagenss2 31045 | . 2 ⊢ ((∪ (𝐽 ×t 𝐽) = ∪ ran 𝑅 ∧ (𝐽 ×t 𝐽) ⊆ (sigaGen‘ran 𝑅) ∧ ran 𝑅 ∈ V) → (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅)) | |
29 | 11, 26, 27, 28 | mp3an 1440 | 1 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∃wrex 3086 Vcvv 3412 ⊆ wss 3828 𝒫 cpw 4420 ∪ cuni 4710 class class class wbr 4927 × cxp 5402 ran crn 5405 ‘cfv 6186 (class class class)co 6974 ∈ cmpo 6976 ωcom 7394 ≼ cdom 8300 ℝcr 10330 1c1 10332 + caddc 10334 / cdiv 11094 2c2 11492 ℤcz 11790 (,)cioo 12551 [,)cico 12553 ↑cexp 13241 topGenctg 16561 Topctop 21199 ×t ctx 21866 sigaGencsigagen 31033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-rep 5047 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 ax-inf2 8894 ax-ac2 9679 ax-cnex 10387 ax-resscn 10388 ax-1cn 10389 ax-icn 10390 ax-addcl 10391 ax-addrcl 10392 ax-mulcl 10393 ax-mulrcl 10394 ax-mulcom 10395 ax-addass 10396 ax-mulass 10397 ax-distr 10398 ax-i2m1 10399 ax-1ne0 10400 ax-1rid 10401 ax-rnegex 10402 ax-rrecex 10403 ax-cnre 10404 ax-pre-lttri 10405 ax-pre-lttrn 10406 ax-pre-ltadd 10407 ax-pre-mulgt0 10408 ax-pre-sup 10409 ax-addf 10410 ax-mulf 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-nel 3071 df-ral 3090 df-rex 3091 df-reu 3092 df-rmo 3093 df-rab 3094 df-v 3414 df-sbc 3681 df-csb 3786 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-pss 3844 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-tp 4444 df-op 4446 df-uni 4711 df-int 4748 df-iun 4792 df-iin 4793 df-br 4928 df-opab 4990 df-mpt 5007 df-tr 5029 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7498 df-2nd 7499 df-supp 7631 df-wrecs 7747 df-recs 7809 df-rdg 7847 df-1o 7901 df-2o 7902 df-oadd 7905 df-omul 7906 df-er 8085 df-map 8204 df-pm 8205 df-ixp 8256 df-en 8303 df-dom 8304 df-sdom 8305 df-fin 8306 df-fsupp 8625 df-fi 8666 df-sup 8697 df-inf 8698 df-oi 8765 df-card 9158 df-acn 9161 df-ac 9332 df-cda 9384 df-pnf 10472 df-mnf 10473 df-xr 10474 df-ltxr 10475 df-le 10476 df-sub 10668 df-neg 10669 df-div 11095 df-nn 11436 df-2 11500 df-3 11501 df-4 11502 df-5 11503 df-6 11504 df-7 11505 df-8 11506 df-9 11507 df-n0 11705 df-z 11791 df-dec 11909 df-uz 12056 df-q 12160 df-rp 12202 df-xneg 12321 df-xadd 12322 df-xmul 12323 df-ioo 12555 df-ioc 12556 df-ico 12557 df-icc 12558 df-fz 12706 df-fzo 12847 df-fl 12974 df-mod 13050 df-seq 13182 df-exp 13242 df-fac 13446 df-bc 13475 df-hash 13503 df-shft 14281 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-limsup 14683 df-clim 14700 df-rlim 14701 df-sum 14898 df-ef 15275 df-sin 15277 df-cos 15278 df-pi 15280 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-refld 20445 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-lp 21442 df-perf 21443 df-cn 21533 df-cnp 21534 df-haus 21621 df-cmp 21693 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-fcls 22247 df-xms 22627 df-ms 22628 df-tms 22629 df-cncf 23183 df-cfil 23555 df-cmet 23557 df-cms 23635 df-limc 24161 df-dv 24162 df-log 24835 df-cxp 24836 df-logb 25038 df-siga 31003 df-sigagen 31034 |
This theorem is referenced by: sxbrsigalem4 31181 |
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