Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 5776 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → ([,) ∘ 𝑖) = ([,) ∘ 𝐼)) | |
3 | 2 | fveq1d 6802 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ 𝐼)‘𝑘)) |
4 | 3 | fveq2d 6804 | . . . . 5 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ 𝐼)‘𝑘))) |
5 | 4 | ralrimivw 3144 | . . . 4 ⊢ (𝑖 = 𝐼 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ 𝐼)‘𝑘))) |
6 | 5 | prodeq2d 15673 | . . 3 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘))) |
7 | hoiprodcl2.i | . . . 4 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) | |
8 | reex 11004 | . . . . . . . 8 ⊢ ℝ ∈ V | |
9 | 8, 8 | xpex 7631 | . . . . . . 7 ⊢ (ℝ × ℝ) ∈ V |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ × ℝ) ∈ V) |
11 | hoiprodcl2.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
12 | 10, 11 | jca 513 | . . . . 5 ⊢ (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin)) |
13 | elmapg 8655 | . . . . 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 258 | . . 3 ⊢ (𝜑 → 𝐼 ∈ ((ℝ × ℝ) ↑m 𝑋)) |
16 | hoiprodcl2.kph | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
17 | 16, 11, 7 | hoiprodcl 44134 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞)) |
18 | 1, 6, 15, 17 | fvmptd3 6926 | . 2 ⊢ (𝜑 → (𝐿‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘))) |
19 | 18, 17 | eqeltrd 2837 | 1 ⊢ (𝜑 → (𝐿‘𝐼) ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 Ⅎwnf 1783 ∈ wcel 2104 Vcvv 3437 ↦ cmpt 5164 × cxp 5594 ∘ ccom 5600 ⟶wf 6450 ‘cfv 6454 (class class class)co 7303 ↑m cmap 8642 Fincfn 8760 ℝcr 10912 0cc0 10913 +∞cpnf 11048 [,)cico 13123 ∏cprod 15656 volcvol 24668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-inf2 9439 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-of 7561 df-om 7741 df-1st 7859 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-2o 8325 df-er 8525 df-map 8644 df-pm 8645 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-fi 9210 df-sup 9241 df-inf 9242 df-oi 9309 df-dju 9699 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-n0 12276 df-z 12362 df-uz 12625 df-q 12731 df-rp 12773 df-xneg 12890 df-xadd 12891 df-xmul 12892 df-ioo 13125 df-ico 13127 df-icc 13128 df-fz 13282 df-fzo 13425 df-fl 13554 df-seq 13764 df-exp 13825 df-hash 14087 df-cj 14851 df-re 14852 df-im 14853 df-sqrt 14987 df-abs 14988 df-clim 15238 df-rlim 15239 df-sum 15439 df-prod 15657 df-rest 17174 df-topgen 17195 df-psmet 20630 df-xmet 20631 df-met 20632 df-bl 20633 df-mopn 20634 df-top 22084 df-topon 22101 df-bases 22137 df-cmp 22579 df-ovol 24669 df-vol 24670 |
This theorem is referenced by: ovnlecvr 44145 ovnsubaddlem1 44157 |
Copyright terms: Public domain | W3C validator |