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Mirrors > Home > MPE Home > Th. List > opnmbl | Structured version Visualization version GIF version |
Description: All open sets are measurable. This proof, via dyadmbl 24204 and uniioombl 24193, shows that it is possible to avoid choice for measurability of open sets and hence continuous functions, which extends the choice-free consequences of Lebesgue measure considerably farther than would otherwise be possible. (Contributed by Mario Carneiro, 26-Mar-2015.) |
Ref | Expression |
---|---|
opnmbl | ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7166 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 / (2↑𝑦)) = (𝑧 / (2↑𝑦))) | |
2 | oveq1 7166 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1)) | |
3 | 2 | oveq1d 7174 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑧 + 1) / (2↑𝑦))) |
4 | 1, 3 | opeq12d 4814 | . . 3 ⊢ (𝑥 = 𝑧 → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝑧 / (2↑𝑦)), ((𝑧 + 1) / (2↑𝑦))〉) |
5 | oveq2 7167 | . . . . 5 ⊢ (𝑦 = 𝑤 → (2↑𝑦) = (2↑𝑤)) | |
6 | 5 | oveq2d 7175 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝑧 / (2↑𝑦)) = (𝑧 / (2↑𝑤))) |
7 | 5 | oveq2d 7175 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 + 1) / (2↑𝑦)) = ((𝑧 + 1) / (2↑𝑤))) |
8 | 6, 7 | opeq12d 4814 | . . 3 ⊢ (𝑦 = 𝑤 → 〈(𝑧 / (2↑𝑦)), ((𝑧 + 1) / (2↑𝑦))〉 = 〈(𝑧 / (2↑𝑤)), ((𝑧 + 1) / (2↑𝑤))〉) |
9 | 4, 8 | cbvmpov 7252 | . 2 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑧 ∈ ℤ, 𝑤 ∈ ℕ0 ↦ 〈(𝑧 / (2↑𝑤)), ((𝑧 + 1) / (2↑𝑤))〉) |
10 | 9 | opnmbllem 24205 | 1 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 〈cop 4576 dom cdm 5558 ran crn 5559 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 1c1 10541 + caddc 10543 / cdiv 11300 2c2 11695 ℕ0cn0 11900 ℤcz 11984 (,)cioo 12741 ↑cexp 13432 topGenctg 16714 volcvol 24067 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-acn 9374 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-rlim 14849 df-sum 15046 df-rest 16699 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-bases 21557 df-cmp 21998 df-ovol 24068 df-vol 24069 |
This theorem is referenced by: subopnmbl 24208 mblfinlem3 34935 mblfinlem4 34936 ismblfin 34937 |
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