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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 34266 | . . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ) |
5 | retop 24798 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | 1, 5 | eqeltri 2835 | . . . 4 ⊢ 𝐽 ∈ Top |
7 | uniretop 24799 | . . . . 5 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
8 | 1 | unieqi 4924 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
9 | 7, 8 | eqtr4i 2766 | . . . 4 ⊢ ℝ = ∪ 𝐽 |
10 | 6, 6, 9, 9 | txunii 23617 | . . 3 ⊢ (ℝ × ℝ) = ∪ (𝐽 ×t 𝐽) |
11 | 4, 10 | eqtr2i 2764 | . 2 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ ran 𝑅 |
12 | 1, 2, 3 | dya2iocuni 34265 | . . . 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 34262 | . . . . . . . . 9 ⊢ ran 𝑅 ≼ ω |
15 | ctex 9003 | . . . . . . . . 9 ⊢ (ran 𝑅 ≼ ω → ran 𝑅 ∈ V) | |
16 | 14, 15 | mp1i 13 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → ran 𝑅 ∈ V) |
17 | elpwi 4612 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → 𝑦 ⊆ ran 𝑅) | |
18 | ssct 9090 | . . . . . . . . 9 ⊢ ((𝑦 ⊆ ran 𝑅 ∧ ran 𝑅 ≼ ω) → 𝑦 ≼ ω) | |
19 | 17, 14, 18 | sylancl 586 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ran 𝑅 → 𝑦 ≼ ω) |
20 | elsigagen2 34129 | . . . . . . . 8 ⊢ ((ran 𝑅 ∈ V ∧ 𝑦 ⊆ ran 𝑅 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ (sigaGen‘ran 𝑅)) | |
21 | 16, 17, 19, 20 | syl3anc 1370 | . . . . . . 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 2840 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 ran 𝑅 ∧ ∪ 𝑦 = 𝑥) → 𝑥 ∈ (sigaGen‘ran 𝑅)) |
24 | 23 | rexlimiva 3145 | . . . 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 3999 | . 2 ⊢ (𝐽 ×t 𝐽) ⊆ (sigaGen‘ran 𝑅) |
27 | 14, 15 | ax-mp 5 | . 2 ⊢ ran 𝑅 ∈ V |
28 | sigagenss2 34131 | . 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 1460 | 1 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 × cxp 5687 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 ≼ cdom 8982 ℝcr 11152 1c1 11154 + caddc 11156 / cdiv 11918 2c2 12319 ℤcz 12611 (,)cioo 13384 [,)cico 13386 ↑cexp 14099 topGenctg 17484 Topctop 22915 ×t ctx 23584 sigaGencsigagen 34119 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-refld 21641 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-fcls 23965 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-cfil 25303 df-cmet 25305 df-cms 25383 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614 df-logb 26823 df-siga 34090 df-sigagen 34120 |
This theorem is referenced by: sxbrsigalem4 34269 |
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