vonsn - Mathbox for Glauco Siliprandi

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Theorem vonsn 47134

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 6827	. . . . 5 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6829	. . . 4 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴}))
3	2	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴}))
4		0fi 8979	. . . . . 6 ⊢ ∅ ∈ Fin
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin)
6		vonsn.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑_m 𝑋))
7	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑_m 𝑋))
8		oveq2 7364	. . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
9	8	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
10	7, 9	eleqtrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑_m ∅))
11	5, 10	snvonmbl 47129	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {𝐴} ∈ dom (voln‘∅))
12	11	von0val 47114	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘{𝐴}) = 0)
13	3, 12	eqtrd 2774	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0)
14		neqne 2942	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
15	14	adantl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
16	6	rrxsnicc 46743	. . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})
17	16	eqcomd 2745	. . . . . 6 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
18	17	fveq2d 6831	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))))
19	18	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))))
20		vonsn.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
21	20	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
22		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
23		elmapi 8786	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑_m 𝑋) → 𝐴:𝑋⟶ℝ)
24	6, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
25	24	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴:𝑋⟶ℝ)
26		eqid 2739	. . . . 5 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))
27	21, 22, 25, 25, 26	vonn0icc 47131	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))))
28	24	ffvelcdmda 7025	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
29	28	rexrd 11186	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
30		iccid 13334	. . . . . . . . . 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 6831	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = (vol‘{(𝐴‘𝑘)}))
33		volsn 46410	. . . . . . . . 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 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0)
36	35	prodeq2dv 15878	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0)
37	36	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0)
38		0cnd 11128	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
39		fprodconst 15934	. . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
40	20, 38, 39	syl2anc 590	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
41	40	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
42		hashnncl 14319	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
43	20, 42	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
44	43	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
45	22, 44	mpbird 258	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
46		0exp 14050	. . . . . 6 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0)
47	45, 46	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (0↑(♯‘𝑋)) = 0)
48	37, 41, 47	3eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0)
49	19, 27, 48	3eqtrd 2778	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = 0)
50	15, 49	syldan 597	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0)
51	13, 50	pm2.61dan 818	1 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∅c0 4261 {csn 4555 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 Xcixp 8835 Fincfn 8883 ℂcc 11027 ℝcr 11028 0cc0 11029 ℝ^*cxr 11169 ℕcn 12165 [,]cicc 13292 ↑cexp 14014 ♯chash 14283 ∏cprod 15859 volcvol 25448 volncvoln 46981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-ac2 10376 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-prod 15860 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-pws 17403 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-drng 20703 df-field 20704 df-abv 20781 df-staf 20811 df-srng 20812 df-lmod 20852 df-lss 20922 df-lmhm 21012 df-lvec 21093 df-sra 21163 df-rgmod 21164 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-cnfld 21348 df-refld 21580 df-phl 21601 df-dsmm 21707 df-frlm 21722 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cn 23210 df-cnp 23211 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-xms 24303 df-ms 24304 df-tms 24305 df-nm 24565 df-ngp 24566 df-tng 24567 df-nrg 24568 df-nlm 24569 df-cncf 24863 df-clm 25048 df-cph 25153 df-tcph 25154 df-rrx 25370 df-ovol 25449 df-vol 25450 df-salg 46752 df-sumge0 46806 df-mea 46893 df-ome 46933 df-caragen 46935 df-ovoln 46980 df-voln 46982

This theorem is referenced by: vonct 47136

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