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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoiprodcl2 | Structured version Visualization version GIF version |
Description: The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
hoiprodcl2.kph | ⊢ Ⅎ𝑘𝜑 |
hoiprodcl2.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
hoiprodcl2.l | ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) |
hoiprodcl2.i | ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) |
Ref | Expression |
---|---|
hoiprodcl2 | ⊢ (𝜑 → (𝐿‘𝐼) ∈ (0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoiprodcl2.l | . . 3 ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) | |
2 | coeq2 5512 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → ([,) ∘ 𝑖) = ([,) ∘ 𝐼)) | |
3 | 2 | fveq1d 6434 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ 𝐼)‘𝑘)) |
4 | 3 | fveq2d 6436 | . . . . 5 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ 𝐼)‘𝑘))) |
5 | 4 | ralrimivw 3175 | . . . 4 ⊢ (𝑖 = 𝐼 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ 𝐼)‘𝑘))) |
6 | 5 | prodeq2d 15024 | . . 3 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘))) |
7 | hoiprodcl2.i | . . . 4 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) | |
8 | reex 10342 | . . . . . . . 8 ⊢ ℝ ∈ V | |
9 | 8, 8 | xpex 7222 | . . . . . . 7 ⊢ (ℝ × ℝ) ∈ V |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ × ℝ) ∈ V) |
11 | hoiprodcl2.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
12 | 10, 11 | jca 509 | . . . . 5 ⊢ (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin)) |
13 | elmapg 8134 | . . . . 5 ⊢ (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → (𝐼 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ 𝐼:𝑋⟶(ℝ × ℝ))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ 𝐼:𝑋⟶(ℝ × ℝ))) |
15 | 7, 14 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝐼 ∈ ((ℝ × ℝ) ↑𝑚 𝑋)) |
16 | hoiprodcl2.kph | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
17 | 16, 11, 7 | hoiprodcl 41554 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞)) |
18 | 1, 6, 15, 17 | fvmptd3 6549 | . 2 ⊢ (𝜑 → (𝐿‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘))) |
19 | 18, 17 | eqeltrd 2905 | 1 ⊢ (𝜑 → (𝐿‘𝐼) ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 Vcvv 3413 ↦ cmpt 4951 × cxp 5339 ∘ ccom 5345 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 ↑𝑚 cmap 8121 Fincfn 8221 ℝcr 10250 0cc0 10251 +∞cpnf 10387 [,)cico 12464 ∏cprod 15007 volcvol 23628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fi 8585 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-ioo 12466 df-ico 12468 df-icc 12469 df-fz 12619 df-fzo 12760 df-fl 12887 df-seq 13095 df-exp 13154 df-hash 13410 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-clim 14595 df-rlim 14596 df-sum 14793 df-prod 15008 df-rest 16435 df-topgen 16456 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-top 21068 df-topon 21085 df-bases 21120 df-cmp 21560 df-ovol 23629 df-vol 23630 |
This theorem is referenced by: ovnlecvr 41565 ovnsubaddlem1 41577 |
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