vonsn - Mathbox for Glauco Siliprandi

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Theorem vonsn 47438

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 6885	. . . . 5 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6887	. . . 4 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴}))
3	2	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴}))
4		0fi 9042	. . . . . 6 ⊢ ∅ ∈ Fin
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin)
6		vonsn.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑_m 𝑋))
7	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑_m 𝑋))
8		oveq2 7424	. . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
9	8	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
10	7, 9	eleqtrd 2867	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑_m ∅))
11	5, 10	snvonmbl 47433	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {𝐴} ∈ dom (voln‘∅))
12	11	von0val 47418	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘{𝐴}) = 0)
13	3, 12	eqtrd 2800	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0)
14		neqne 2968	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
15	14	adantl 487	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
16	6	rrxsnicc 47047	. . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})
17	16	eqcomd 2771	. . . . . 6 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
18	17	fveq2d 6889	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))))
19	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))))
20		vonsn.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
21	20	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
22		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
23		elmapi 8848	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑_m 𝑋) → 𝐴:𝑋⟶ℝ)
24	6, 23	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
25	24	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴:𝑋⟶ℝ)
26		eqid 2765	. . . . 5 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))
27	21, 22, 25, 25, 26	vonn0icc 47435	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))))
28	24	ffvelcdmda 7083	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
29	28	rexrd 11270	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
30		iccid 13428	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘) ∈ ℝ^* → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {(𝐴‘𝑘)})
31	29, 30	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {(𝐴‘𝑘)})
32	31	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = (vol‘{(𝐴‘𝑘)}))
33		volsn 46714	. . . . . . . . 9 ⊢ ((𝐴‘𝑘) ∈ ℝ → (vol‘{(𝐴‘𝑘)}) = 0)
34	28, 33	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘{(𝐴‘𝑘)}) = 0)
35	32, 34	eqtrd 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0)
36	35	prodeq2dv 15994	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0)
37	36	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0)
38		0cnd 11210	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
39		fprodconst 16050	. . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
40	20, 38, 39	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
41	40	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
42		hashnncl 14415	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
43	20, 42	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
44	43	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
45	22, 44	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
46		0exp 14146	. . . . . 6 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0)
47	45, 46	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (0↑(♯‘𝑋)) = 0)
48	37, 41, 47	3eqtrd 2804	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0)
49	19, 27, 48	3eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = 0)
50	15, 49	syldan 603	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0)
51	13, 50	pm2.61dan 825	1 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∅c0 4286 {csn 4591 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 ↑_m cmap 8826 Xcixp 8897 Fincfn 8945 ℂcc 11109 ℝcr 11110 0cc0 11111 ℝ^*cxr 11253 ℕcn 12244 [,]cicc 13386 ↑cexp 14110 ♯chash 14379 ∏cprod 15975 volcvol 25651 volncvoln 47285

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cc 10430 ax-ac2 10458 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190 ax-mulf 11191

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-dju 9899 df-card 9937 df-acn 9940 df-ac 10112 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12723 df-uz 12874 df-q 12984 df-rp 13028 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13547 df-fzo 13695 df-fl 13838 df-seq 14051 df-exp 14111 df-hash 14380 df-cj 15169 df-re 15170 df-im 15171 df-sqrt 15305 df-abs 15306 df-clim 15558 df-rlim 15559 df-sum 15757 df-prod 15976 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-starv 17342 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ds 17349 df-unif 17350 df-hom 17351 df-cco 17352 df-rest 17492 df-topn 17493 df-0g 17511 df-gsum 17512 df-topgen 17513 df-pt 17514 df-prds 17517 df-pws 17519 df-xrs 17573 df-qtop 17578 df-imas 17579 df-xps 17581 df-mre 17655 df-mrc 17656 df-acs 17658 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-mhm 18864 df-submnd 18865 df-grp 19026 df-minusg 19027 df-sbg 19028 df-mulg 19157 df-subg 19212 df-ghm 19307 df-cntz 19410 df-cmn 19875 df-abl 19876 df-mgp 20240 df-rng 20254 df-ur 20287 df-ring 20340 df-cring 20341 df-oppr 20444 df-dvdsr 20464 df-unit 20465 df-invr 20495 df-dvr 20508 df-rhm 20579 df-subrng 20674 df-subrg 20698 df-drng 20858 df-field 20859 df-abv 20941 df-staf 20971 df-srng 20972 df-lmod 21012 df-lss 21082 df-lmhm 21172 df-lvec 21253 df-sra 21323 df-rgmod 21324 df-psmet 21543 df-xmet 21544 df-met 21545 df-bl 21546 df-mopn 21547 df-cnfld 21552 df-refld 21784 df-phl 21805 df-dsmm 21911 df-frlm 21926 df-top 23080 df-topon 23097 df-topsp 23119 df-bases 23132 df-cn 23413 df-cnp 23414 df-cmp 23573 df-tx 23748 df-hmeo 23941 df-xms 24506 df-ms 24507 df-tms 24508 df-nm 24768 df-ngp 24769 df-tng 24770 df-nrg 24771 df-nlm 24772 df-cncf 25066 df-clm 25251 df-cph 25356 df-tcph 25357 df-rrx 25573 df-ovol 25652 df-vol 25653 df-salg 47056 df-sumge0 47110 df-mea 47197 df-ome 47237 df-caragen 47239 df-ovoln 47284 df-voln 47286

This theorem is used by: vonct 47440

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