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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 25649 and uniioombl 25638, 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 7438 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 / (2↑𝑦)) = (𝑧 / (2↑𝑦))) | |
2 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1)) | |
3 | 2 | oveq1d 7446 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑧 + 1) / (2↑𝑦))) |
4 | 1, 3 | opeq12d 4886 | . . 3 ⊢ (𝑥 = 𝑧 → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝑧 / (2↑𝑦)), ((𝑧 + 1) / (2↑𝑦))〉) |
5 | oveq2 7439 | . . . . 5 ⊢ (𝑦 = 𝑤 → (2↑𝑦) = (2↑𝑤)) | |
6 | 5 | oveq2d 7447 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝑧 / (2↑𝑦)) = (𝑧 / (2↑𝑤))) |
7 | 5 | oveq2d 7447 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 + 1) / (2↑𝑦)) = ((𝑧 + 1) / (2↑𝑤))) |
8 | 6, 7 | opeq12d 4886 | . . 3 ⊢ (𝑦 = 𝑤 → 〈(𝑧 / (2↑𝑦)), ((𝑧 + 1) / (2↑𝑦))〉 = 〈(𝑧 / (2↑𝑤)), ((𝑧 + 1) / (2↑𝑤))〉) |
9 | 4, 8 | cbvmpov 7528 | . 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 25650 | 1 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 〈cop 4637 dom cdm 5689 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1c1 11154 + caddc 11156 / cdiv 11918 2c2 12319 ℕ0cn0 12524 ℤcz 12611 (,)cioo 13384 ↑cexp 14099 topGenctg 17484 volcvol 25512 |
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-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 |
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-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 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-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-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 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-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cmp 23411 df-ovol 25513 df-vol 25514 |
This theorem is referenced by: subopnmbl 25653 mblfinlem3 37646 mblfinlem4 37647 ismblfin 37648 |
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