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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 1544 | . . 3 ⊢ ⊤ | |
2 | 0fin 9177 | . . . . . 6 ⊢ ∅ ∈ Fin | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ Fin) |
4 | 3 | ovnf 45738 | . . . 4 ⊢ (⊤ → (voln*‘∅):𝒫 (ℝ ↑m ∅)⟶(0[,]+∞)) |
5 | 4 | feqmptd 6960 | . . 3 ⊢ (⊤ → (voln*‘∅) = (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥))) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 (ℝ ↑m ∅) ↦ ((voln*‘∅)‘𝑥)) |
7 | reex 11207 | . . . . 5 ⊢ ℝ ∈ V | |
8 | mapdm0 8842 | . . . . 5 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅}) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (ℝ ↑m ∅) = {∅} |
10 | 9 | pweqi 4618 | . . 3 ⊢ 𝒫 (ℝ ↑m ∅) = 𝒫 {∅} |
11 | mpteq1 5241 | . . 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 4609 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 {∅} → 𝑥 ⊆ {∅}) | |
14 | 9 | eqcomi 2740 | . . . . . 6 ⊢ {∅} = (ℝ ↑m ∅) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 {∅} → {∅} = (ℝ ↑m ∅)) |
16 | 13, 15 | sseqtrd 4022 | . . . 4 ⊢ (𝑥 ∈ 𝒫 {∅} → 𝑥 ⊆ (ℝ ↑m ∅)) |
17 | 16 | ovn0val 45725 | . . 3 ⊢ (𝑥 ∈ 𝒫 {∅} → ((voln*‘∅)‘𝑥) = 0) |
18 | 17 | mpteq2ia 5251 | . 2 ⊢ (𝑥 ∈ 𝒫 {∅} ↦ ((voln*‘∅)‘𝑥)) = (𝑥 ∈ 𝒫 {∅} ↦ 0) |
19 | 6, 12, 18 | 3eqtri 2763 | 1 ⊢ (voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2105 Vcvv 3473 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8826 Fincfn 8945 ℝcr 11115 0cc0 11116 +∞cpnf 11252 [,]cicc 13334 voln*covoln 45711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-prod 15857 df-rest 17375 df-topgen 17396 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-top 22716 df-topon 22733 df-bases 22769 df-cmp 23211 df-ovol 25313 df-vol 25314 df-sumge0 45538 df-ovoln 45712 |
This theorem is referenced by: (None) |
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