Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > vonn0icc | Structured version Visualization version GIF version |
Description: The n-dimensional Lebesgue measure of a closed interval, when the dimension of the space is nonzero. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
vonn0icc.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
vonn0icc.n | ⊢ (𝜑 → 𝑋 ≠ ∅) |
vonn0icc.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
vonn0icc.b | ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
vonn0icc.i | ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)) |
Ref | Expression |
---|---|
vonn0icc | ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐵‘𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonn0icc.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | vonn0icc.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
3 | vonn0icc.b | . . . 4 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
4 | vonn0icc.i | . . . 4 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)) | |
5 | fveq2 6103 | . . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑎‘𝑗) = (𝑎‘𝑘)) | |
6 | fveq2 6103 | . . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑏‘𝑗) = (𝑏‘𝑘)) | |
7 | 5, 6 | oveq12d 6567 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → ((𝑎‘𝑗)[,)(𝑏‘𝑗)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
8 | 7 | fveq2d 6107 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))) = (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) |
9 | 8 | cbvprodv 14485 | . . . . . . . 8 ⊢ ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
10 | ifeq2 4041 | . . . . . . . 8 ⊢ (∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) → if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ (ℝ ↑𝑚 𝑥) ∧ 𝑏 ∈ (ℝ ↑𝑚 𝑥)) → if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) |
13 | 12 | mpt2eq3ia 6618 | . . . . 5 ⊢ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))))) = (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) |
14 | 13 | mpteq2i 4669 | . . . 4 ⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗)))))) = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
15 | 1, 2, 3, 4, 14 | vonicc 39576 | . . 3 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))))))‘𝑋)𝐵)) |
16 | 14 | fveq1i 6104 | . . . . 5 ⊢ ((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))))))‘𝑋) = ((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))‘𝑋) |
17 | 16 | oveqi 6562 | . . . 4 ⊢ (𝐴((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))))))‘𝑋)𝐵) = (𝐴((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))‘𝑋)𝐵) |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑗 ∈ 𝑥 (vol‘((𝑎‘𝑗)[,)(𝑏‘𝑗))))))‘𝑋)𝐵) = (𝐴((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))‘𝑋)𝐵)) |
19 | 15, 18 | eqtrd 2644 | . 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))‘𝑋)𝐵)) |
20 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) | |
21 | vonn0icc.n | . . 3 ⊢ (𝜑 → 𝑋 ≠ ∅) | |
22 | 20, 1, 21, 2, 3 | hoidmvn0val 39474 | . 2 ⊢ (𝜑 → (𝐴((𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
23 | 2 | ffvelrnda 6267 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
24 | 3 | ffvelrnda 6267 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
25 | 23, 24 | voliccico 38892 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐵‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
26 | 25 | eqcomd 2616 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑘)[,](𝐵‘𝑘)))) |
27 | 26 | prodeq2dv 14492 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐵‘𝑘)))) |
28 | 19, 22, 27 | 3eqtrd 2648 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐵‘𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ifcif 4036 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑𝑚 cmap 7744 Xcixp 7794 Fincfn 7841 ℝcr 9814 0cc0 9815 [,)cico 12048 [,]cicc 12049 ∏cprod 14474 volcvol 23039 volncvoln 39428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-prod 14475 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-pws 15933 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-field 18573 df-subrg 18601 df-abv 18640 df-staf 18668 df-srng 18669 df-lmod 18688 df-lss 18754 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-refld 19770 df-phl 19790 df-dsmm 19895 df-frlm 19910 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-tng 22199 df-nrg 22200 df-nlm 22201 df-cncf 22489 df-clm 22671 df-cph 22776 df-tch 22777 df-rrx 22981 df-ovol 23040 df-vol 23041 df-salg 39205 df-sumge0 39256 df-mea 39343 df-ome 39380 df-caragen 39382 df-ovoln 39427 df-voln 39429 |
This theorem is referenced by: vonsn 39582 vonn0icc2 39583 |
Copyright terms: Public domain | W3C validator |