vonsn - Mathbox for Glauco Siliprandi

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Theorem vonsn 46662

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.)

**Hypotheses**
Ref	Expression
vonsn.1	⊢ (𝜑 → 𝑋 ∈ Fin)
vonsn.2	⊢ (𝜑 → 𝐴 ∈ (ℝ ↑_m 𝑋))

**Assertion**
Ref	Expression
vonsn	⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0)

Proof of Theorem vonsn Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6840	. . . . 5 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6842	. . . 4 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴}))
3	2	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴}))
4		0fi 8990	. . . . . 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 7377	. . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
9	8	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
10	7, 9	eleqtrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑_m ∅))
11	5, 10	snvonmbl 46657	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {𝐴} ∈ dom (voln‘∅))
12	11	von0val 46642	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘{𝐴}) = 0)
13	3, 12	eqtrd 2764	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0)
14		neqne 2933	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
15	14	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
16	6	rrxsnicc 46271	. . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})
17	16	eqcomd 2735	. . . . . 6 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
18	17	fveq2d 6844	. . . . 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 8799	. . . . . . 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 2729	. . . . 5 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))
27	21, 22, 25, 25, 26	vonn0icc 46659	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))))
28	24	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
29	28	rexrd 11200	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
30		iccid 13327	. . . . . . . . . 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 6844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = (vol‘{(𝐴‘𝑘)}))
33		volsn 45938	. . . . . . . . 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 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0)
36	35	prodeq2dv 15864	. . . . . 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 11143	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
39		fprodconst 15920	. . . . . . 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 14307	. . . . . . . . 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 14038	. . . . . 6 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0)
47	45, 46	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (0↑(♯‘𝑋)) = 0)
48	37, 41, 47	3eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0)
49	19, 27, 48	3eqtrd 2768	. . 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 1540 ∈ wcel 2109 ≠ wne 2925 ∅c0 4292 {csn 4585 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ↑_m cmap 8776 Xcixp 8847 Fincfn 8895 ℂcc 11042 ℝcr 11043 0cc0 11044 ℝ^*cxr 11183 ℕcn 12162 [,]cicc 13285 ↑cexp 14002 ♯chash 14271 ∏cprod 15845 volcvol 25340 volncvoln 46509

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-ac2 10392 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-acn 9871 df-ac 10045 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-prod 15846 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-pws 17388 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-mulg 18976 df-subg 19031 df-ghm 19121 df-cntz 19225 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-drng 20616 df-field 20617 df-abv 20694 df-staf 20724 df-srng 20725 df-lmod 20744 df-lss 20814 df-lmhm 20905 df-lvec 20986 df-sra 21056 df-rgmod 21057 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-refld 21490 df-phl 21511 df-dsmm 21617 df-frlm 21632 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cn 23090 df-cnp 23091 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-xms 24184 df-ms 24185 df-tms 24186 df-nm 24446 df-ngp 24447 df-tng 24448 df-nrg 24449 df-nlm 24450 df-cncf 24747 df-clm 24939 df-cph 25044 df-tcph 25045 df-rrx 25261 df-ovol 25341 df-vol 25342 df-salg 46280 df-sumge0 46334 df-mea 46421 df-ome 46461 df-caragen 46463 df-ovoln 46508 df-voln 46510

This theorem is referenced by: vonct 46664

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