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 1537 | . . 3 ⊢ ⊤ | |
2 | 0fin 8740 | . . . . . 6 ⊢ ∅ ∈ Fin | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ Fin) |
4 | 3 | ovnf 42838 | . . . 4 ⊢ (⊤ → (voln*‘∅):𝒫 (ℝ ↑m ∅)⟶(0[,]+∞)) |
5 | 4 | feqmptd 6728 | . . 3 ⊢ (⊤ → (voln*‘∅) = (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥))) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥)) |
7 | reex 10622 | . . . . 5 ⊢ ℝ ∈ V | |
8 | mapdm0 8415 | . . . . 5 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅}) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (ℝ ↑m ∅) = {∅} |
10 | 9 | pweqi 4543 | . . 3 ⊢ 𝒫 (ℝ ↑m ∅) = 𝒫 {∅} |
11 | mpteq1 5147 | . . 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 4551 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 {∅} → 𝑥 ⊆ {∅}) | |
14 | 9 | eqcomi 2830 | . . . . . 6 ⊢ {∅} = (ℝ ↑m ∅) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 {∅} → {∅} = (ℝ ↑m ∅)) |
16 | 13, 15 | sseqtrd 4007 | . . . 4 ⊢ (𝑥 ∈ 𝒫 {∅} → 𝑥 ⊆ (ℝ ↑m ∅)) |
17 | 16 | ovn0val 42825 | . . 3 ⊢ (𝑥 ∈ 𝒫 {∅} → ((voln*‘∅)‘𝑥) = 0) |
18 | 17 | mpteq2ia 5150 | . 2 ⊢ (𝑥 ∈ 𝒫 {∅} ↦ ((voln*‘∅)‘𝑥)) = (𝑥 ∈ 𝒫 {∅} ↦ 0) |
19 | 6, 12, 18 | 3eqtri 2848 | 1 ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 𝒫 cpw 4539 {csn 4561 ↦ cmpt 5139 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 ℝcr 10530 0cc0 10531 +∞cpnf 10666 [,]cicc 12735 voln*covoln 42811 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-prod 15254 df-rest 16690 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cmp 21989 df-ovol 24059 df-vol 24060 df-sumge0 42638 df-ovoln 42812 |
This theorem is referenced by: (None) |
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