Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovn02 | Structured version Visualization version GIF version |
Description: For the zero-dimensional space, voln* assigns zero to every subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ovn02 | ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1547 | . . 3 ⊢ ⊤ | |
2 | 0fin 8849 | . . . . . 6 ⊢ ∅ ∈ Fin | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ Fin) |
4 | 3 | ovnf 43776 | . . . 4 ⊢ (⊤ → (voln*‘∅):𝒫 (ℝ ↑m ∅)⟶(0[,]+∞)) |
5 | 4 | feqmptd 6780 | . . 3 ⊢ (⊤ → (voln*‘∅) = (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥))) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥)) |
7 | reex 10820 | . . . . 5 ⊢ ℝ ∈ V | |
8 | mapdm0 8523 | . . . . 5 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅}) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (ℝ ↑m ∅) = {∅} |
10 | 9 | pweqi 4531 | . . 3 ⊢ 𝒫 (ℝ ↑m ∅) = 𝒫 {∅} |
11 | mpteq1 5143 | . . 3 ⊢ (𝒫 (ℝ ↑m ∅) = 𝒫 {∅} → (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥)) = (𝑥 ∈ 𝒫 {∅} ↦ ((voln*‘∅)‘𝑥))) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥)) = (𝑥 ∈ 𝒫 {∅} ↦ ((voln*‘∅)‘𝑥)) |
13 | elpwi 4522 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 {∅} → 𝑥 ⊆ {∅}) | |
14 | 9 | eqcomi 2746 | . . . . . 6 ⊢ {∅} = (ℝ ↑m ∅) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 {∅} → {∅} = (ℝ ↑m ∅)) |
16 | 13, 15 | sseqtrd 3941 | . . . 4 ⊢ (𝑥 ∈ 𝒫 {∅} → 𝑥 ⊆ (ℝ ↑m ∅)) |
17 | 16 | ovn0val 43763 | . . 3 ⊢ (𝑥 ∈ 𝒫 {∅} → ((voln*‘∅)‘𝑥) = 0) |
18 | 17 | mpteq2ia 5146 | . 2 ⊢ (𝑥 ∈ 𝒫 {∅} ↦ ((voln*‘∅)‘𝑥)) = (𝑥 ∈ 𝒫 {∅} ↦ 0) |
19 | 6, 12, 18 | 3eqtri 2769 | 1 ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 Vcvv 3408 ∅c0 4237 𝒫 cpw 4513 {csn 4541 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 Fincfn 8626 ℝcr 10728 0cc0 10729 +∞cpnf 10864 [,]cicc 12938 voln*covoln 43749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-prod 15468 df-rest 16927 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-bases 21843 df-cmp 22284 df-ovol 24361 df-vol 24362 df-sumge0 43576 df-ovoln 43750 |
This theorem is referenced by: (None) |
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