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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidmv0val | Structured version Visualization version GIF version |
Description: The dimensional volume of a 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
hoidmv0val.l | ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
hoidmv0val.a | ⊢ (𝜑 → 𝐴:∅⟶ℝ) |
hoidmv0val.b | ⊢ (𝜑 → 𝐵:∅⟶ℝ) |
Ref | Expression |
---|---|
hoidmv0val | ⊢ (𝜑 → (𝐴(𝐿‘∅)𝐵) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoidmv0val.l | . . 3 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) | |
2 | hoidmv0val.a | . . 3 ⊢ (𝜑 → 𝐴:∅⟶ℝ) | |
3 | hoidmv0val.b | . . 3 ⊢ (𝜑 → 𝐵:∅⟶ℝ) | |
4 | 0fin 9167 | . . . 4 ⊢ ∅ ∈ Fin | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ Fin) |
6 | 1, 2, 3, 5 | hoidmvval 45228 | . 2 ⊢ (𝜑 → (𝐴(𝐿‘∅)𝐵) = if(∅ = ∅, 0, ∏𝑘 ∈ ∅ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))) |
7 | eqid 2733 | . . . 4 ⊢ ∅ = ∅ | |
8 | iftrue 4533 | . . . 4 ⊢ (∅ = ∅ → if(∅ = ∅, 0, ∏𝑘 ∈ ∅ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) = 0) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ if(∅ = ∅, 0, ∏𝑘 ∈ ∅ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) = 0 |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → if(∅ = ∅, 0, ∏𝑘 ∈ ∅ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) = 0) |
11 | 6, 10 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐴(𝐿‘∅)𝐵) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∅c0 4321 ifcif 4527 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 ↑m cmap 8816 Fincfn 8935 ℝcr 11105 0cc0 11106 [,)cico 13322 ∏cprod 15845 volcvol 24962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-seq 13963 df-prod 15846 |
This theorem is referenced by: hoidmvval0b 45241 hoidmvlelem2 45247 hoidmvlelem3 45248 hoidmvle 45251 ovnhoi 45254 vonioo 45333 vonicc 45336 |
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