Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > hoimbl | Structured version Visualization version GIF version |
Description: Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
hoimbl.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
hoimbl.s | ⊢ 𝑆 = dom (voln‘𝑋) |
hoimbl.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
hoimbl.b | ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
Ref | Expression |
---|---|
hoimbl | ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoimbl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 ∈ Fin) |
3 | 2 | rrnmbl 40828 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑𝑚 𝑋) ∈ dom (voln‘𝑋)) |
4 | reex 10027 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
5 | mapdm0 7872 | . . . . . . . . 9 ⊢ (ℝ ∈ V → (ℝ ↑𝑚 ∅) = {∅}) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (ℝ ↑𝑚 ∅) = {∅} |
7 | 6 | eqcomi 2631 | . . . . . . 7 ⊢ {∅} = (ℝ ↑𝑚 ∅) |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑋 = ∅ → {∅} = (ℝ ↑𝑚 ∅)) |
9 | id 22 | . . . . . . . 8 ⊢ (𝑋 = ∅ → 𝑋 = ∅) | |
10 | 9 | ixpeq1d 7920 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖))) |
11 | ixp0x 7936 | . . . . . . . 8 ⊢ X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅} | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
13 | 10, 12 | eqtrd 2656 | . . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
14 | oveq2 6658 | . . . . . 6 ⊢ (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅)) | |
15 | 8, 13, 14 | 3eqtr4d 2666 | . . . . 5 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = (ℝ ↑𝑚 𝑋)) |
16 | 15 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = (ℝ ↑𝑚 𝑋)) |
17 | hoimbl.s | . . . . 5 ⊢ 𝑆 = dom (voln‘𝑋) | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑆 = dom (voln‘𝑋)) |
19 | 16, 18 | eleq12d 2695 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆 ↔ (ℝ ↑𝑚 𝑋) ∈ dom (voln‘𝑋))) |
20 | 3, 19 | mpbird 247 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
21 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
22 | 9 | necon3bi 2820 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) |
23 | 22 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
24 | hoimbl.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
25 | 24 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ) |
26 | hoimbl.b | . . . 4 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
27 | 26 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ) |
28 | id 22 | . . . . . 6 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥) | |
29 | eqidd 2623 | . . . . . 6 ⊢ (𝑤 = 𝑥 → ℝ = ℝ) | |
30 | 28 | ixpeq1d 7920 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) |
31 | eqeq1 2626 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 = ℎ ↔ 𝑖 = ℎ)) | |
32 | 31 | ifbid 4108 | . . . . . . . . 9 ⊢ (𝑗 = 𝑖 → if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
33 | 32 | cbvixpv 7926 | . . . . . . . 8 ⊢ X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) |
34 | 33 | a1i 11 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
35 | 30, 34 | eqtrd 2656 | . . . . . 6 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
36 | 28, 29, 35 | mpt2eq123dv 6717 | . . . . 5 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) = (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ))) |
37 | eqeq2 2633 | . . . . . . . . 9 ⊢ (ℎ = 𝑙 → (𝑖 = ℎ ↔ 𝑖 = 𝑙)) | |
38 | 37 | ifbid 4108 | . . . . . . . 8 ⊢ (ℎ = 𝑙 → if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
39 | 38 | ixpeq2dv 7924 | . . . . . . 7 ⊢ (ℎ = 𝑙 → X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
40 | oveq2 6658 | . . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (-∞(,)𝑧) = (-∞(,)𝑦)) | |
41 | 40 | ifeq1d 4104 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
42 | 41 | ixpeq2dv 7924 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
43 | 39, 42 | cbvmpt2v 6735 | . . . . . 6 ⊢ (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
44 | 43 | a1i 11 | . . . . 5 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
45 | 36, 44 | eqtrd 2656 | . . . 4 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
46 | 45 | cbvmptv 4750 | . . 3 ⊢ (𝑤 ∈ Fin ↦ (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ))) = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
47 | 21, 23, 17, 25, 27, 46 | hoimbllem 40844 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
48 | 20, 47 | pm2.61dan 832 | 1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 ifcif 4086 {csn 4177 ↦ cmpt 4729 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ↑𝑚 cmap 7857 Xcixp 7908 Fincfn 7955 ℝcr 9935 -∞cmnf 10072 (,)cioo 12175 [,)cico 12177 volncvoln 40752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-prod 14636 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-bases 20750 df-cmp 21190 df-ovol 23233 df-vol 23234 df-salg 40529 df-sumge0 40580 df-mea 40667 df-ome 40704 df-caragen 40706 df-ovoln 40751 df-voln 40753 |
This theorem is referenced by: opnvonmbllem2 40847 hoimbl2 40879 vonhoi 40881 vonioolem1 40894 vonioolem2 40895 vonicclem1 40897 vonicclem2 40898 |
Copyright terms: Public domain | W3C validator |