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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonsn | Structured version Visualization version GIF version |
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 6907 | . . . . 5 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅)) | |
2 | 1 | fveq1d 6909 | . . . 4 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) |
4 | 0fi 9081 | . . . . . 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 7439 | . . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) | |
9 | 8 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) |
10 | 7, 9 | eleqtrd 2841 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑m ∅)) |
11 | 5, 10 | snvonmbl 46642 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {𝐴} ∈ dom (voln‘∅)) |
12 | 11 | von0val 46627 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘{𝐴}) = 0) |
13 | 3, 12 | eqtrd 2775 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
14 | neqne 2946 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
15 | 14 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
16 | 6 | rrxsnicc 46256 | . . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴}) |
17 | 16 | eqcomd 2741 | . . . . . 6 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
18 | 17 | fveq2d 6911 | . . . . 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 8888 | . . . . . . 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 2735 | . . . . 5 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) | |
27 | 21, 22, 25, 25, 26 | vonn0icc 46644 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
28 | 24 | ffvelcdmda 7104 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
29 | 28 | rexrd 11309 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*) |
30 | iccid 13429 | . . . . . . . . . 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 6911 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = (vol‘{(𝐴‘𝑘)})) |
33 | volsn 45923 | . . . . . . . . 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 2775 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) |
36 | 35 | prodeq2dv 15955 | . . . . . 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 11252 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ) | |
39 | fprodconst 16011 | . . . . . . 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 14402 | . . . . . . . . 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 14135 | . . . . . 6 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0) | |
47 | 45, 46 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (0↑(♯‘𝑋)) = 0) |
48 | 37, 41, 47 | 3eqtrd 2779 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) |
49 | 19, 27, 48 | 3eqtrd 2779 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
50 | 15, 49 | syldan 591 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
51 | 13, 50 | pm2.61dan 813 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 {csn 4631 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Xcixp 8936 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 ℝ*cxr 11292 ℕcn 12264 [,]cicc 13387 ↑cexp 14099 ♯chash 14366 ∏cprod 15936 volcvol 25512 volncvoln 46494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-prod 15937 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-pws 17496 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-abv 20827 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-refld 21641 df-phl 21662 df-dsmm 21770 df-frlm 21785 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-cnp 23252 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-tng 24613 df-nrg 24614 df-nlm 24615 df-cncf 24918 df-clm 25110 df-cph 25216 df-tcph 25217 df-rrx 25433 df-ovol 25513 df-vol 25514 df-salg 46265 df-sumge0 46319 df-mea 46406 df-ome 46446 df-caragen 46448 df-ovoln 46493 df-voln 46495 |
This theorem is referenced by: vonct 46649 |
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