Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonvol2 | Structured version Visualization version GIF version |
Description: The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
vonvol2.f | ⊢ Ⅎ𝑓𝑌 |
vonvol2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
vonvol2.x | ⊢ (𝜑 → 𝑋 ∈ dom (voln‘{𝐴})) |
vonvol2.y | ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 |
Ref | Expression |
---|---|
vonvol2 | ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = (vol‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonvol2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | vonvol2.f | . . . . . . 7 ⊢ Ⅎ𝑓𝑌 | |
3 | snfi 8038 | . . . . . . . . 9 ⊢ {𝐴} ∈ Fin | |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → {𝐴} ∈ Fin) |
5 | vonvol2.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom (voln‘{𝐴})) | |
6 | 4, 5 | vonmblss2 40856 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑𝑚 {𝐴})) |
7 | vonvol2.y | . . . . . . 7 ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 | |
8 | 2, 1, 6, 7 | ssmapsn 39408 | . . . . . 6 ⊢ (𝜑 → 𝑋 = (𝑌 ↑𝑚 {𝐴})) |
9 | 8 | eqcomd 2628 | . . . . 5 ⊢ (𝜑 → (𝑌 ↑𝑚 {𝐴}) = 𝑋) |
10 | 9, 5 | eqeltrd 2701 | . . . 4 ⊢ (𝜑 → (𝑌 ↑𝑚 {𝐴}) ∈ dom (voln‘{𝐴})) |
11 | 6 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (ℝ ↑𝑚 {𝐴})) |
12 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋) | |
13 | 11, 12 | sseldd 3604 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑𝑚 {𝐴})) |
14 | elmapi 7879 | . . . . . . . . 9 ⊢ (𝑓 ∈ (ℝ ↑𝑚 {𝐴}) → 𝑓:{𝐴}⟶ℝ) | |
15 | frn 6053 | . . . . . . . . 9 ⊢ (𝑓:{𝐴}⟶ℝ → ran 𝑓 ⊆ ℝ) | |
16 | 13, 14, 15 | 3syl 18 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ) |
17 | 16 | ralrimiva 2966 | . . . . . . 7 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
18 | iunss 4561 | . . . . . . 7 ⊢ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ ↔ ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) | |
19 | 17, 18 | sylibr 224 | . . . . . 6 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
20 | 7, 19 | syl5eqss 3649 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
21 | 1, 20 | vonvolmbl 40875 | . . . 4 ⊢ (𝜑 → ((𝑌 ↑𝑚 {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol)) |
22 | 10, 21 | mpbid 222 | . . 3 ⊢ (𝜑 → 𝑌 ∈ dom vol) |
23 | 1, 22 | vonvol 40876 | . 2 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑌 ↑𝑚 {𝐴})) = (vol‘𝑌)) |
24 | 9 | eqcomd 2628 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑌 ↑𝑚 {𝐴})) |
25 | 24 | fveq2d 6195 | . 2 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = ((voln‘{𝐴})‘(𝑌 ↑𝑚 {𝐴}))) |
26 | eqidd 2623 | . 2 ⊢ (𝜑 → (vol‘𝑌) = (vol‘𝑌)) | |
27 | 23, 25, 26 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = (vol‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ⊆ wss 3574 {csn 4177 ∪ ciun 4520 dom cdm 5114 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Fincfn 7955 ℝcr 9935 volcvol 23232 volncvoln 40752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-prod 14636 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-bases 20750 df-cmp 21190 df-ovol 23233 df-vol 23234 df-sumge0 40580 df-ome 40704 df-caragen 40706 df-ovoln 40751 df-voln 40753 |
This theorem is referenced by: (None) |
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