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| Description: The n-dimensional Lebesgue measure of a singleton is zero. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| vonsn.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) | 
| vonsn.2 | ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋)) | 
| Ref | Expression | 
|---|---|
| vonsn | ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . . . 5 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅)) | |
| 2 | 1 | fveq1d 6907 | . . . 4 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) | 
| 3 | 2 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) | 
| 4 | 0fi 9083 | . . . . . 6 ⊢ ∅ ∈ Fin | |
| 5 | 4 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin) | 
| 6 | vonsn.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋)) | |
| 7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑m 𝑋)) | 
| 8 | oveq2 7440 | . . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) | |
| 9 | 8 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) | 
| 10 | 7, 9 | eleqtrd 2842 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑m ∅)) | 
| 11 | 5, 10 | snvonmbl 46706 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {𝐴} ∈ dom (voln‘∅)) | 
| 12 | 11 | von0val 46691 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘{𝐴}) = 0) | 
| 13 | 3, 12 | eqtrd 2776 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) | 
| 14 | neqne 2947 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
| 15 | 14 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) | 
| 16 | 6 | rrxsnicc 46320 | . . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴}) | 
| 17 | 16 | eqcomd 2742 | . . . . . 6 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) | 
| 18 | 17 | fveq2d 6909 | . . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))) | 
| 19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))) | 
| 20 | vonsn.1 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
| 21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin) | 
| 22 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
| 23 | elmapi 8890 | . . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → 𝐴:𝑋⟶ℝ) | |
| 24 | 6, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | 
| 25 | 24 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴:𝑋⟶ℝ) | 
| 26 | eqid 2736 | . . . . 5 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) | |
| 27 | 21, 22, 25, 25, 26 | vonn0icc 46708 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘)))) | 
| 28 | 24 | ffvelcdmda 7103 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) | 
| 29 | 28 | rexrd 11312 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*) | 
| 30 | iccid 13433 | . . . . . . . . . 10 ⊢ ((𝐴‘𝑘) ∈ ℝ* → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {(𝐴‘𝑘)}) | |
| 31 | 29, 30 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {(𝐴‘𝑘)}) | 
| 32 | 31 | fveq2d 6909 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = (vol‘{(𝐴‘𝑘)})) | 
| 33 | volsn 45987 | . . . . . . . . 9 ⊢ ((𝐴‘𝑘) ∈ ℝ → (vol‘{(𝐴‘𝑘)}) = 0) | |
| 34 | 28, 33 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘{(𝐴‘𝑘)}) = 0) | 
| 35 | 32, 34 | eqtrd 2776 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) | 
| 36 | 35 | prodeq2dv 15959 | . . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0) | 
| 37 | 36 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0) | 
| 38 | 0cnd 11255 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ) | |
| 39 | fprodconst 16015 | . . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋))) | |
| 40 | 20, 38, 39 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋))) | 
| 41 | 40 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋))) | 
| 42 | hashnncl 14406 | . . . . . . . . 9 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) | |
| 43 | 20, 42 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) | 
| 44 | 43 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) | 
| 45 | 22, 44 | mpbird 257 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ) | 
| 46 | 0exp 14139 | . . . . . 6 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0) | |
| 47 | 45, 46 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (0↑(♯‘𝑋)) = 0) | 
| 48 | 37, 41, 47 | 3eqtrd 2780 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) | 
| 49 | 19, 27, 48 | 3eqtrd 2780 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = 0) | 
| 50 | 15, 49 | syldan 591 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) | 
| 51 | 13, 50 | pm2.61dan 812 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 {csn 4625 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 Xcixp 8938 Fincfn 8986 ℂcc 11154 ℝcr 11155 0cc0 11156 ℝ*cxr 11295 ℕcn 12267 [,]cicc 13391 ↑cexp 14103 ♯chash 14370 ∏cprod 15940 volcvol 25499 volncvoln 46558 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-acn 9983 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-rlim 15526 df-sum 15724 df-prod 15941 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 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 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-pws 17495 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-drng 20732 df-field 20733 df-abv 20811 df-staf 20841 df-srng 20842 df-lmod 20861 df-lss 20931 df-lmhm 21022 df-lvec 21103 df-sra 21173 df-rgmod 21174 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-cnfld 21366 df-refld 21624 df-phl 21645 df-dsmm 21753 df-frlm 21768 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cn 23236 df-cnp 23237 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-xms 24331 df-ms 24332 df-tms 24333 df-nm 24596 df-ngp 24597 df-tng 24598 df-nrg 24599 df-nlm 24600 df-cncf 24905 df-clm 25097 df-cph 25203 df-tcph 25204 df-rrx 25420 df-ovol 25500 df-vol 25501 df-salg 46329 df-sumge0 46383 df-mea 46470 df-ome 46510 df-caragen 46512 df-ovoln 46557 df-voln 46559 | 
| This theorem is referenced by: vonct 46713 | 
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