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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 34286 | . . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ) |
| 5 | retop 24782 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
| 6 | 1, 5 | eqeltri 2837 | . . . 4 ⊢ 𝐽 ∈ Top |
| 7 | uniretop 24783 | . . . . 5 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 8 | 1 | unieqi 4919 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
| 9 | 7, 8 | eqtr4i 2768 | . . . 4 ⊢ ℝ = ∪ 𝐽 |
| 10 | 6, 6, 9, 9 | txunii 23601 | . . 3 ⊢ (ℝ × ℝ) = ∪ (𝐽 ×t 𝐽) |
| 11 | 4, 10 | eqtr2i 2766 | . 2 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ ran 𝑅 |
| 12 | 1, 2, 3 | dya2iocuni 34285 | . . . 4 ⊢ (𝑥 ∈ (𝐽 ×t 𝐽) → ∃𝑦 ∈ 𝒫 ran 𝑅∪ 𝑦 = 𝑥) |
| 13 | simpr 484 | . . . . . 6 ⊢ ((𝑦 ∈ 𝒫 ran 𝑅 ∧ ∪ 𝑦 = 𝑥) → ∪ 𝑦 = 𝑥) | |
| 14 | 1, 2, 3 | dya2iocct 34282 | . . . . . . . . 9 ⊢ ran 𝑅 ≼ ω |
| 15 | ctex 9004 | . . . . . . . . 9 ⊢ (ran 𝑅 ≼ ω → ran 𝑅 ∈ V) | |
| 16 | 14, 15 | mp1i 13 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → ran 𝑅 ∈ V) |
| 17 | elpwi 4607 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → 𝑦 ⊆ ran 𝑅) | |
| 18 | ssct 9091 | . . . . . . . . 9 ⊢ ((𝑦 ⊆ ran 𝑅 ∧ ran 𝑅 ≼ ω) → 𝑦 ≼ ω) | |
| 19 | 17, 14, 18 | sylancl 586 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → 𝑦 ≼ ω) |
| 20 | elsigagen2 34149 | . . . . . . . 8 ⊢ ((ran 𝑅 ∈ V ∧ 𝑦 ⊆ ran 𝑅 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ (sigaGen‘ran 𝑅)) | |
| 21 | 16, 17, 19, 20 | syl3anc 1373 | . . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → ∪ 𝑦 ∈ (sigaGen‘ran 𝑅)) |
| 22 | 21 | adantr 480 | . . . . . 6 ⊢ ((𝑦 ∈ 𝒫 ran 𝑅 ∧ ∪ 𝑦 = 𝑥) → ∪ 𝑦 ∈ (sigaGen‘ran 𝑅)) |
| 23 | 13, 22 | eqeltrrd 2842 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 ran 𝑅 ∧ ∪ 𝑦 = 𝑥) → 𝑥 ∈ (sigaGen‘ran 𝑅)) |
| 24 | 23 | rexlimiva 3147 | . . . 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 3987 | . 2 ⊢ (𝐽 ×t 𝐽) ⊆ (sigaGen‘ran 𝑅) |
| 27 | 14, 15 | ax-mp 5 | . 2 ⊢ ran 𝑅 ∈ V |
| 28 | sigagenss2 34151 | . 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 1463 | 1 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 × cxp 5683 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 ≼ cdom 8983 ℝcr 11154 1c1 11156 + caddc 11158 / cdiv 11920 2c2 12321 ℤcz 12613 (,)cioo 13387 [,)cico 13389 ↑cexp 14102 topGenctg 17482 Topctop 22899 ×t ctx 23568 sigaGencsigagen 34139 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-refld 21623 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-fcls 23949 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-cfil 25289 df-cmet 25291 df-cms 25369 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 df-logb 26808 df-siga 34110 df-sigagen 34140 |
| This theorem is referenced by: sxbrsigalem4 34289 |
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