Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccvonmbllem | Structured version Visualization version GIF version |
Description: Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
iccvonmbllem.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
iccvonmbllem.s | ⊢ 𝑆 = dom (voln‘𝑋) |
iccvonmbllem.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
iccvonmbllem.b | ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
iccvonmbllem.c | ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛)))) |
iccvonmbllem.d | ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛)))) |
Ref | Expression |
---|---|
iccvonmbllem | ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccvonmbllem.c | . . . . . . . . . . . 12 ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛)))) | |
2 | 1 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))))) |
3 | iccvonmbllem.x | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
4 | 3 | adantr 481 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) |
5 | 4 | mptexd 6487 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))) ∈ V) |
6 | 2, 5 | fvmpt2d 6293 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛)))) |
7 | iccvonmbllem.a | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
8 | 7 | ffvelrnda 6359 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ) |
9 | 8 | adantlr 751 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ) |
10 | nnrecre 11057 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
11 | 10 | ad2antlr 763 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
12 | 9, 11 | resubcld 10458 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖) − (1 / 𝑛)) ∈ ℝ) |
13 | 6, 12 | fvmpt2d 6293 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑖) = ((𝐴‘𝑖) − (1 / 𝑛))) |
14 | 13 | an32s 846 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛)‘𝑖) = ((𝐴‘𝑖) − (1 / 𝑛))) |
15 | iccvonmbllem.d | . . . . . . . . . . . 12 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛)))) | |
16 | 15 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))))) |
17 | 4 | mptexd 6487 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))) ∈ V) |
18 | 16, 17 | fvmpt2d 6293 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛)))) |
19 | iccvonmbllem.b | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
20 | 19 | ffvelrnda 6359 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ) |
21 | 20 | adantlr 751 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ) |
22 | 21, 11 | readdcld 10069 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) + (1 / 𝑛)) ∈ ℝ) |
23 | 18, 22 | fvmpt2d 6293 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑛)‘𝑖) = ((𝐵‘𝑖) + (1 / 𝑛))) |
24 | 23 | an32s 846 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)‘𝑖) = ((𝐵‘𝑖) + (1 / 𝑛))) |
25 | 14, 24 | oveq12d 6668 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛)))) |
26 | 25 | iineq2dv 4543 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ∩ 𝑛 ∈ ℕ (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛)))) |
27 | 8, 20 | iooiinicc 39769 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛))) = ((𝐴‘𝑖)[,](𝐵‘𝑖))) |
28 | 26, 27 | eqtrd 2656 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ((𝐴‘𝑖)[,](𝐵‘𝑖))) |
29 | 28 | ixpeq2dva 7923 | . . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖))) |
30 | 29 | eqcomd 2628 | . . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖))) |
31 | eqidd 2623 | . . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖))) | |
32 | nnn0 39595 | . . . . 5 ⊢ ℕ ≠ ∅ | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ ≠ ∅) |
34 | ixpiin 7934 | . . . 4 ⊢ (ℕ ≠ ∅ → X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖))) | |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖))) |
36 | 30, 31, 35 | 3eqtr3d 2664 | . 2 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖))) |
37 | iccvonmbllem.s | . . . 4 ⊢ 𝑆 = dom (voln‘𝑋) | |
38 | 3, 37 | dmovnsal 40826 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
39 | nnct 12780 | . . . 4 ⊢ ℕ ≼ ω | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
41 | eqid 2622 | . . . . . . 7 ⊢ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))) = (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))) | |
42 | 12, 41 | fmptd 6385 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ) |
43 | ressxr 10083 | . . . . . . 7 ⊢ ℝ ⊆ ℝ* | |
44 | 43 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℝ ⊆ ℝ*) |
45 | 42, 44 | fssd 6057 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ*) |
46 | 6 | feq1d 6030 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ* ↔ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ*)) |
47 | 45, 46 | mpbird 247 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ*) |
48 | eqid 2622 | . . . . . . 7 ⊢ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))) = (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))) | |
49 | 22, 48 | fmptd 6385 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ) |
50 | 49, 44 | fssd 6057 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ*) |
51 | 18 | feq1d 6030 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛):𝑋⟶ℝ* ↔ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ*)) |
52 | 50, 51 | mpbird 247 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛):𝑋⟶ℝ*) |
53 | 4, 37, 47, 52 | ioovonmbl 40891 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) ∈ 𝑆) |
54 | 38, 40, 33, 53 | saliincl 40545 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) ∈ 𝑆) |
55 | 36, 54 | eqeltrd 2701 | 1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ∩ ciin 4521 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ωcom 7065 Xcixp 7908 ≼ cdom 7953 Fincfn 7955 ℝcr 9935 1c1 9937 + caddc 9939 ℝ*cxr 10073 − cmin 10266 / cdiv 10684 ℕcn 11020 (,)cioo 12175 [,]cicc 12178 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-supp 7296 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-fsupp 8276 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-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-prds 16108 df-pws 16110 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-ghm 17658 df-cntz 17750 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-rnghom 18715 df-drng 18749 df-field 18750 df-subrg 18778 df-abv 18817 df-staf 18845 df-srng 18846 df-lmod 18865 df-lss 18933 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-refld 19951 df-phl 19971 df-dsmm 20076 df-frlm 20091 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cmp 21190 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 df-tng 22389 df-nrg 22390 df-nlm 22391 df-clm 22863 df-cph 22968 df-tch 22969 df-rrx 23173 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: iccvonmbl 40893 |
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