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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonvolmbl2 | Structured version Visualization version GIF version |
Description: A subset 𝑋 of the space of 1-dimensional Real numbers is Lebesgue measurable if and only if its projection 𝑌 on the Real numbers is Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
vonvolmbl2.f | ⊢ Ⅎ𝑓𝑌 |
vonvolmbl2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
vonvolmbl2.x | ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑m {𝐴})) |
vonvolmbl2.y | ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 |
Ref | Expression |
---|---|
vonvolmbl2 | ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonvolmbl2.f | . . . 4 ⊢ Ⅎ𝑓𝑌 | |
2 | vonvolmbl2.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | vonvolmbl2.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑m {𝐴})) | |
4 | vonvolmbl2.y | . . . 4 ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 | |
5 | 1, 2, 3, 4 | ssmapsn 41499 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑌 ↑m {𝐴})) |
6 | 5 | eleq1d 2897 | . 2 ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ (𝑌 ↑m {𝐴}) ∈ dom (voln‘{𝐴}))) |
7 | 3 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (ℝ ↑m {𝐴})) |
8 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋) | |
9 | 7, 8 | sseldd 3968 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑m {𝐴})) |
10 | elmapi 8428 | . . . . . . 7 ⊢ (𝑓 ∈ (ℝ ↑m {𝐴}) → 𝑓:{𝐴}⟶ℝ) | |
11 | frn 6520 | . . . . . . 7 ⊢ (𝑓:{𝐴}⟶ℝ → ran 𝑓 ⊆ ℝ) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ) |
13 | 12 | ralrimiva 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
14 | iunss 4969 | . . . . 5 ⊢ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ ↔ ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) | |
15 | 13, 14 | sylibr 236 | . . . 4 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
16 | 4, 15 | eqsstrid 4015 | . . 3 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
17 | 2, 16 | vonvolmbl 42963 | . 2 ⊢ (𝜑 → ((𝑌 ↑m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol)) |
18 | 6, 17 | bitrd 281 | 1 ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 ∀wral 3138 ⊆ wss 3936 {csn 4567 ∪ ciun 4919 dom cdm 5555 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℝcr 10536 volcvol 24064 volncvoln 42840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-prod 15260 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-0g 16715 df-topgen 16717 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-bases 21554 df-cmp 21995 df-ovol 24065 df-vol 24066 df-sumge0 42665 df-ome 42792 df-caragen 42794 df-ovoln 42839 df-voln 42841 |
This theorem is referenced by: (None) |
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